cuspidalizations.dvi

ABSOLUTE ANABELIAN CUSPIDALIZATIONS OF PROPER HYPERBOLIC CURVES Shinichi Mochizuki June 2007 In this paper, we develop the theory of “cuspidalizations” of the étale fundamental group of a proper hyperbolic curve over a ﬁnite or nonarchimedean mixed-characteristic local ﬁeld. The ultimate goal of this theory is the group-theoretic reconstruction of the étale fundamental group of an arbitrary open subscheme of the curve from the étale fundamental group of the full proper curve. We then apply this theory to show that a certain absolute p-adic version of the Grothendieck Conjec- ture holds for hyperbolic curves “of Belyi type”. This includes, in particular, aﬃne hyperbolic curves over a nonarchimedean mixed-characteristic local ﬁeld which are deﬁned over a number ﬁeld and isogenous to a hyperbolic curve of genus zero. Also, we apply this theory to prove the analogue for proper hyperbolic curves over ﬁnite ﬁelds of the version of the Grothendieck Conjecture that was shown in [Tama]. Contents: §0. Notations and Conventions §1. Maximal Abelian Cuspidalizations §2. Points and Functions §3. Maximal Pro-l Cuspidalizations Appendix: Free Lie Algebras Introduction Let X be a proper hyperbolic curve over a ﬁeld k which is either ﬁnite or nonarchimedean local of mixed characteristic; let U ⊆ X be an open subscheme of X. Write Π X for the étale fundamental group of X. In this paper, we study the extent to which the étale fundamental group of U may be group-theoretically reconstructed from Π X . In §1, we show that the abelian portion of the extension of Π X determined by the étale fundamental group of U may be group-theoretically reconstructed from Π X [cf. Theorem 1.16, (iii)], and, moreover, that this construction has certain remarkable rigidity properties [cf. Propositions 1.15, (i); 2.6, (i)]. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI In §2, we show that this abelian portion of the extension is suﬃcient to recon- struct [in essence] the multiplicative group of the function ﬁeld of X [cf. Theorem 2.5, (ii)]. In the case of nonarchimedean [mixed-characteristic] local ﬁelds, this already implies various interesting consequences in the context of the absolute an- abelian geometry studied in [Mzk5], [Mzk6], [Mzk8]. In particular, it implies that the absolute p-adic version of the Grothendieck Conjecture [i.e., an absolute version of [a certain portion of] the relative result that appears as the main result of [Mzk4]] holds for hyperbolic curves “of Belyi type” [cf. Deﬁnition 2.9; Corollary 2.12]. This includes, in particular, hyperbolic curves “of strictly Belyi type”, i.e., aﬃne hy- perbolic curves over a nonarchimedean [mixed-characteristic] local ﬁeld which are deﬁned over a number ﬁeld and isogenous to a hyperbolic curve of genus zero. In particular, we obtain a new countable class of “absolute curves” [in the terminology of [Mzk6]], whose absoluteness is, in certain respects, reminiscent of the absolute- ness of the canonical curves of p-adic Teichmüller theory discussed in [Mzk6] [cf. Remark 2.13.1], but [in contrast to the class of canonical curves] appears [at least from the point of view of certain circumstantial evidence] unlikely to be Zariski dense in most moduli spaces [cf. Remark 2.13.2]. Finally, in §3, we apply the theory of the weight ﬁltration [cf., e.g., [Kane], [Mtm]], together with various generalities concerning free Lie algebras [cf. the Appendix], to develop various “higher order generalizations” of the theory of §1, 2. In particular, we obtain various “higher order generalizations” of the “remarkable rigidity” referred to above [cf. Propositions 3.7, 3.9, especially Proposition 3.9, (iii)], which we apply to show that, relative to the notation introduced above, the geometrically pro-l portion [where l is a prime number invertible in k] of the étale fundamental group of U may be recovered from Π X , at least when U is obtained from X by removing a single k-rational point [cf. Theorem 3.10]. This, along with the theory of §2, allows one to verify the analogue for proper hyperbolic curves over ﬁnite ﬁelds of the version of the Grothendieck Conjecture that was shown in [Tama] [cf. Theorem 3.12]. Acknowledgements: I would like to thank Akio Tamagawa, Makoto Matsumoto, and Seidai Yasuda for various useful comments. Also, I would like to thank Yuichiro Hoshi for his careful reading of an earlier version of this manuscript, which led to the discovery of various errors in this earlier version. ABSOLUTE ANABELIAN CUSPIDALIZATIONS 3 Section 0: Notations and Conventions Numbers: the proﬁnite completion of the additive group of rational We shall denote by Z integers Z. If p is a prime number, then Z p denotes the ring of p-adic integers; Q p denotes its quotient ﬁeld. We shall refer to as a p-adic local ﬁeld (respectively, nonarchimedean local ﬁeld) any ﬁnite ﬁeld extension of Q p (respectively, a p-adic local ﬁeld, for some p). A number ﬁeld is deﬁned to be a ﬁnite extension of the ﬁeld of rational numbers. If Σ is a set of prime numbers, then we shall refer to a positive integer each of whose prime factors belongs to Σ as a Σ-integer. We shall refer to a ﬁnite étale covering of schemes whose degree is a Σ-integer as a Σ-covering. Also, we shall write Primes for the set of all prime numbers. Topological Groups: Let G be a Hausdorﬀ topological group, and H ⊆ G a closed subgroup. Let us write G ab for the abelianization of G [i.e., the quotient of G by the closed subgroup of G topologically generated by the commutators of G]. Let us write def Z G (H) = {g ∈ G | g · h = h · g, ∀ h ∈ H} for the centralizer of H in G; N G (H) = {g ∈ G | g · H · g −1 = H} def for the normalizer of H in G; and def C G (H) = {g ∈ G | (g · H · g −1 ) H has ﬁnite index in H, g · H · g −1 } for the commensurator of H in G. Note that: (i) Z G (H), N G (H) and C G (H) are subgroups of G; (ii) we have inclusions H, Z G (H) ⊆ N G (H) ⊆ C G (H) and (iii) H is normal in N G (H). If H = N G (H) (respectively, H = C G (H)), then we shall say that H is normally terminal (respectively, commensurably terminal) in G. Note that Z G (H), N G (H) are always closed in G, while C G (H) is not necessarily closed in G. If G 1 , G 2 are Hausdorﬀ topological groups, then an outer homomorphism G 1 → G 2 is deﬁned to be an equivalence class of continuous homomorphisms G 1 → G 2 , where two such homomorphisms are considered equivalent if they diﬀer 4 SHINICHI MOCHIZUKI by composition with an inner automorphism of G 2 . The group of outer automor- phisms of G [i.e., bijective bicontinuous outer homomorphisms G → G] will be denoted Out(G). If G is center-free, then there is a natural exact sequence: 1 → G → Aut(G) → Out(G) → 1 [where the homomorphism G → Aut(G) is deﬁned by letting G act on G by conju- gation]. If G is a proﬁnite group such that, for every open subgroup H ⊆ G, we have Z G (H) = {1}, then we shall say that G is slim. One veriﬁes immediately that G is slim if and only if every open subgroup of G is center-free [cf. [Mzk5], Remark 0.1.3]. If G is a proﬁnite group and Σ is set of prime numbers, then we shall say that G is a pro-Σ group if the order of every ﬁnite quotient group of G is a Σ-integer. If Σ = {l} is of cardinality one, then we shall refer to a pro-Σ group as a pro-l group. Curves: Suppose that g ≥ 0 is an integer. Then if S is a scheme, a family of curves of genus g X → S is deﬁned to be a smooth, proper, geometrically connected morphism of schemes X → S whose geometric ﬁbers are curves of genus g. Suppose that g, r ≥ 0 are integers such that 2g − 2 + r > 0. We shall denote the moduli stack of r-pointed stable curves of genus g (where we assume the points to be unordered) by M g,r [cf. [DM], [Knud] for an exposition of the theory of such curves; strictly speaking, [Knud] treats the ﬁnite étale covering of M g,r determined by ordering the marked points]. The open substack M g,r ⊆ M g,r of smooth curves will be referred to as the moduli stack of smooth r-pointed stable curves of genus g or, alternatively, as the moduli stack of hyperbolic curves of type (g, r). A family of hyperbolic curves of type (g, r) X → S is deﬁned to be a morphism which factors X → Y → S as the composite of an open immersion X → Y onto the complement Y \D of a relative divisor D ⊆ Y which is ﬁnite étale over S of relative degree r, and a family Y → S of curves of genus g. One checks easily that, if S is normal, then the pair (Y, D) is unique up to canonical isomorphism. (Indeed, when S is the spectrum of a ﬁeld, this fact is well-known from the elementary theory of algebraic curves. Next, we consider an arbitrary connected normal S on which a prime l is invertible (which, by Zariski localization, we may assume without loss of generality). Denote by S → S the ﬁnite étale covering parametrizing orderings of the marked points and trivializations of the l-torsion points of the Jacobian of Y . Note that S → S is independent of ABSOLUTE ANABELIAN CUSPIDALIZATIONS 5 the choice of (Y, D), since (by the normality of S), S may be constructed as the normalization of S in the function ﬁeld of S (which is independent of the choice of (Y, D) since the restriction of (Y, D) to the generic point of S has already been shown to be unique). Thus, the uniqueness of (Y, D) follows by considering the classifying morphism (associated to (Y, D)) from S to the ﬁnite étale covering of (M g,r ) Z[ 1 l ] parametrizing orderings of the marked points and trivializations of the l-torsion points of the Jacobian [since this covering is well-known to be a scheme, for l suﬃciently large].) We shall refer to Y (respectively, D; D) as the compactiﬁcation (respectively, divisor of cusps; divisor of marked points) of X. A family of hyperbolic curves X → S is deﬁned to be a morphism X → S such that the restriction of this morphism to each connected component of S is a family of hyperbolic curves of type (g, r) for some integers (g, r) as above. A family of hyperbolic curves X → S of type (0, 3) will be referred to as a tripod. If X is a hyperbolic curve over a ﬁeld K with compactiﬁcation X ⊆ X, then we shall write X cl ; X cl+ for the sets of closed points of X and X, respectively. If X K (respectively, Y L ) is a hyperbolic curve over a ﬁeld K (respectively, L), then we shall say that X K is isogenous to Y L if there exists a hyperbolic curve Z M over a ﬁeld M together with ﬁnite étale morphisms Z M → X K , Z M → Y L . Note that in this situation, the morphisms Z M → X K , Z M → Y L induce ﬁnite separable inclusions of ﬁelds K → M , L → M . [Indeed, this follows immediately from the × easily veriﬁed fact that every subgroup G ⊆ Γ(Z, O Z ) such that G {0} determines a ﬁeld is necessarily contained in M × .] If X is a generically scheme-like algebraic stack [i.e., an algebraic stack which admits a “scheme-theoretically” dense open that is isomorphic to a scheme] over a ﬁeld K of characteristic zero that admits a [surjective] ﬁnite étale [or, equivalently, ﬁnite étale Galois] covering Y → X, where Y is a hyperbolic curve over a ﬁnite extension of K, then we shall refer to X as a hyperbolic orbicurve over K. [Although this deﬁnition diﬀers from the deﬁnition of a “hyperbolic orbicurve” given in [Mzk6], Deﬁnition 2.2, (ii), it follows immediately from a theorem of Bundgaard-Nielsen-Fox [cf., e.g., [Namba], Theorem 1.2.15, p. 29] that these two deﬁnitions are equivalent.] If X → Y is a dominant morphism of hyperbolic orbicurves, then we shall refer to X → Y as a partial coarsiﬁcation morphism if the morphism induced by X → Y on associated coarse spaces [cf., e.g., [FC], Chapter I, §4.10] is an isomorphism. Let X be a hyperbolic orbicurve over an algebraically closed ﬁeld of character- istic zero; denote its étale fundamental group by Δ X . We shall refer to the order of the [manifestly ﬁnite!] decomposition group of a closed point x of X as the order of x. We shall refer to the [manifestly ﬁnite!] least common multiple of the orders of the closed points of X as the order of X. Thus, it follows immediately from the deﬁnitions that X is a hyperbolic curve if and only if the order of X is equal to 1. 6 SHINICHI MOCHIZUKI Section 1: Maximal Abelian Cuspidalizations Let X be a proper hyperbolic curve over a ﬁeld k which is either ﬁnite or nonarchimedean local. Write d k for the cohomological dimension of k. Thus, if k is ﬁnite (respectively, nonar- chimedean local), then d k = 1 (respectively, d k = 2 [cf., e.g., [NSW], Chapter 7, Theorem 7.1.8, (i)]). If k is ﬁnite (respectively, nonarchimedean local), we shall denote the characteristic of k (respectively, of the residue ﬁeld of k) by p and the number p (respectively, 1) by p † . Also, we shall write Primes † = Primes\(Primes def {p † }) [where Primes is the set of all prime numbers [cf. §0]; the intersection is taken in the “ambient set” Z]. Let Σ be a set of prime numbers that contains at least one prime number that is invertible in k. Write Σ = Σ\(Σ def {p}); Σ † = Σ\(Σ def {p † }) the max- [where the intersections are taken in the “ambient set” Z]. Denote by Z and by Z † the maximal pro-Σ † quotient of Z. imal pro-Σ quotient of Z If k is an algebraic closure of k, then we shall denote the result of base-changing objects over k to k by means of a subscript “k”. Any choice of a basepoint of X determines an algebraic closure k of k, and hence an exact sequence 1 → π 1 (X k ) → π 1 (X) → G k → 1 def where G k = Gal(k/k); π 1 (X), π 1 (X k ) are the étale fundamental groups of X, def X k , respectively. Write Δ X for the maximal pro-Σ quotient of π 1 (X k ) and Π X = π 1 (X)/Ker(π 1 (X k ) Δ X ). Thus, we have an exact sequence: 1 → Δ X → Π X → G k → 1 def Similarly, if we write X × X = X × k X, then we obtain [by considering the maximal pro-Σ quotient of π 1 ((X × X) k )] an exact sequence 1 → Δ X×X → Π X×X → G k → 1 where Π X×X (respectively, Δ X×X ) may be identiﬁed with Π X × G k Π X (respectively, Δ X × Δ X ). Let Π Z ⊆ Π X×X be an open subgroup that surjects onto G k . Write def Z → X × X for the corresponding covering; Δ Z = Ker(Π Z G k ). ABSOLUTE ANABELIAN CUSPIDALIZATIONS 7 † A Proposition 1.1. (Group-theoreticity of Étale Cohomology) Let Z be a ﬁnite quotient, and N a ﬁnite A-module equipped with a continuous Δ X - (respectively, Π X -; Δ Z -; Π Z -) action. Then for i ∈ Z, the natural homomorphism i H i (Δ X , N ) → H ét (X k , N ) i (respectively, H i (Π X , N ) → H ét (X, N ); i (Z k , N ); H i (Δ Z , N ) → H ét i H i (Π Z , N ) → H ét (Z, N )) is an isomorphism. Proof. In light of the Leray spectral sequence for the surjections Π X G k , Π Z Im(Π Z ) ⊆ Π X [i.e., where “Im(−)” denotes the image via the natural homomorphism associated to one of the projections Z → X × X → X], it suﬃces to verify the asserted isomorphism in the case of Δ X . If Y → X k is a connected ﬁnite étale Galois Σ-covering, then the associated Leray spectral sequence has “E 2 -term” given by the cohomology of Gal(Y /X) with coeﬃcients in the étale cohomology of Y and abuts to the étale cohomology of X k . By allowing Y to vary, one then veriﬁes immediately that it suﬃces to verify that every étale cohomology class of Y [with coeﬃcients in N ] vanishes upon pull-back to some [connected] ﬁnite étale Σ-covering Y → Y . Moreover, by passing to an appropriate Y , one reduces immediately to the case where N = A, equipped with the trivial Π X -action. Then the vanishing assertion in question is a tautology for “H 1 ”; for “H 2 ”, it suﬃces to take Y → Y so that the degree of Y → Y annihilates A [cf., e.g., the discussion at the bottom of [FK], p. 136]. Set: † ), Z † ); M X = Hom Z † (H 2 (Δ X , Z def def ⊗d −1 M k = Hom Z † (H d k (G k , M X k ⊗d −1 ), M X k ) † -modules of rank one; M X is isomorphic as a G k -module Thus, M k , M X are free Z † (1) via the † (1) [where the “(1)” denotes a “Tate twist” — i.e., G k acts on Z to Z † (d −1). [Indeed, this cyclotomic character]; M k is isomorphic as a G k -module to Z k follows from Proposition 1.1; Poincaré duality [cf., e.g., [FK], Chapter II, Theorem [together with an easy compu- 1.13]; the fact, in the ﬁnite ﬁeld case, that G k ∼ = Z the well-known theory of the cohomology of tation of the group cohomology of Z]; nonarchimedean local ﬁelds [cf., e.g., [NSW], Chapter 7, Theorem 7.2.6].] Remark 1.2.0. Note that for any open subgroup Π X ⊆ Π X [which we think of as corresponding to a ﬁnite étale covering X → X], we obtain a natural isomorphism ∼ M X → M X † ) to the induced morphism on group coho- by applying the functor Hom Z † (−, Z † ) → H 2 (Δ X , Z † ) [where Δ X def = Ker(Π X → G k )] and dividing mology H 2 (Δ X , Z by [Δ X : Δ X ]. [One veriﬁes easily that this does indeed yield an isomorphism — cf., e.g., the discussion at the bottom of [FK], p. 136.] Moreover, relative to 8 SHINICHI MOCHIZUKI † , H 2 (Δ X , M X ) ∼ † , the iso- the tautological isomorphisms H 2 (Δ X , M X ) ∼ = Z = Z ∼ morphism M X → M X just constructed induces [via the restriction morphism on † given by multiplication by [Δ X : Δ X ]. † → Z group cohomology] the morphism Z Similarly, if k is the base ﬁeld of X , then we obtain a natural isomorphism ∼ M k → M k ∼ by applying the natural isomorphism M X → M X just constructed and the dual of the natural pull-back morphism on group cohomology and then dividing by [k : k] [cf., e.g., [NSW], Chapter 7, Corollary 7.1.4]. Proposition 1.2. (Top Cohomology Modules) (i) There are natural isomorphisms: † ; H d k (G k , M k ) ∼ = Z † ; H 2 (Δ X , M X ) ∼ = Z ⊗2 ∼ † H 4 (Δ Z , M X ) = Z ; † H d k +2 (Π X , M X ⊗ M k ) ∼ = Z ⊗2 † H d k +4 (Π Z , M X ⊗ M k ) ∼ = Z ∼ † (ii) There is a unique isomorphism M X → Z (1) such that the image of † † ∼ 1 ∈ Z maps via the composite of the isomorphism Z = H 2 (Δ X , M X ) of (i) ∼ † (1)) induced by the isomorphism with the isomorphism H 2 (Δ X , M X ) → H 2 (Δ X , Z ∼ † M X → Z (1) in question to the [ﬁrst] Chern class of a line bundle of degree 1 on X k . Proof. Assertion (i) follows from the deﬁnitions; the Leray spectral sequence for the surjections Π X G k , Π Z Im(Π Z ) ⊆ Π X [i.e., where “Im(−)” denotes the image via the natural homomorphism associated to one of the projections Z → X × X → X]. Assertion (ii) is immediate from the deﬁnitions. † A be a ﬁnite quotient, and Proposition 1.3. (Duality) For i ∈ Z, let Z N a ﬁnite A-module. (i) Suppose that N is equipped with a continuous G k -action. Then the pairing H i (G k , N ) × H d k −i (G k , Hom A (N, M k ⊗ A)) → A determined by the cup product in group cohomology and the natural isomorphisms of Proposition 1.2, (i), determines an isomorphism as follows: ∼ H i (G k , N ) → Hom A (H d k −i (G k , Hom A (N, M k ⊗ A)), A) (ii) Suppose that N is equipped with a continuous Π X - (respectively, Δ X -; Π Z -; Δ Z -) action. Then the pairing ABSOLUTE ANABELIAN CUSPIDALIZATIONS 9 H i (Π X , N ) × H d k +2−i (Π X , Hom A (N, M X ⊗ M k ⊗ A)) → A (respectively, H i (Δ X , N ) × H 2−i (Δ X , Hom A (N, M X ⊗ A)) → A; ⊗2 ⊗ M k ⊗ A)) → A; H i (Π Z , N ) × H d k +4−i (Π Z , Hom A (N, M X ⊗2 H i (Δ Z , N ) × H 4−i (Δ Z , Hom A (N, M X ⊗ A)) → A) determined by the cup product in group cohomology and the natural isomorphisms of Proposition 1.2, (i), determines an isomorphism as follows: ∼ H i (Π X , N ) → Hom A (H d k +2−i (Π X , Hom A (N, M X ⊗ M k ⊗ A)), A) ∼ (respectively, H i (Δ X , N ) → Hom A (H 2−i (Δ X , Hom A (N, M X ⊗ A)), A); ∼ ⊗2 ⊗ M k ⊗ A)), A); H i (Π Z , N ) → Hom A (H d k +4−i (Π Z , Hom A (N, M X ∼ ⊗2 ⊗ A)), A)) H i (Δ Z , N ) → Hom A (H 4−i (Δ Z , Hom A (N, M X [together with Proof. Assertion (i) follows immediately from the fact that G k ∼ = Z in the ﬁnite ﬁeld case; [NSW], an easy computation of the group cohomology of Z] Chapter 7, Theorem 7.2.6, in the nonarchimedean local ﬁeld case. Assertion (ii) then follows from assertion (i); the Leray spectral sequences associated to Π X G k , Π Z Im(Π Z ) ⊆ Π X [i.e., where “Im(−)” denotes the image via the natural homomorphism associated to one of the projections Z → X × X → X]; Proposition 1.1; Poincaré duality [cf., e.g., [FK], Chapter II, Theorem 1.13]. Proposition 1.4. (Automorphisms of Cyclotomic Extensions) (i) We have: H 0 (G k , H 1 (Δ X , M X )) = 0. (ii) There are natural isomorphisms ∼ ∼ ∧ ∼ ∼ ∧ H 1 (Π X , M X ) → H 1 (G k , M X ) → (k × ) H 1 (Π Z , M X ) → H 1 (G k , M X ) → (k × ) — where the ﬁrst isomorphisms in each line are induced by the surjections Π X G k , Π Z G k ; the second isomorphisms in each line are induced by the isomor- ∧ phism of Proposition 1.2, (ii), and the Kummer exact sequence; (k × ) is the max- imal pro-Σ † -quotient of k × . Proof. Assertion (i) follows immediately from the “Riemann hypothesis for abelian varieties over ﬁnite ﬁelds” [cf., e.g., [Mumf], p. 206] in the ﬁnite ﬁeld case; [Mzk8], Lemma 4.6, in the nonarchimedean local ﬁeld case. The ﬁrst isomorphisms of assertion (ii) follow immediately from assertion (i) and the Leray spectral sequences 10 SHINICHI MOCHIZUKI associated to Π X G k , Π Z G k ; the second isomorphisms follow immediately from consideration of the Kummer exact sequence for Spec(k). Deﬁnition 1.5. (i) Let H be a proﬁnite group equipped with a homomorphism H → Π X . Then we shall refer to the kernel I H of H → Π X as the cuspidal subgroup of H [relative to H → Π X ]. We shall say that H is cuspidally abelian (respectively, cuspidally pro-Σ ∗ [where Σ ∗ is a set of prime numbers]) [relative to H → Π X ] if I H is abelian (respectively, a pro-Σ ∗ group). If H is cuspidally abelian, then observe that H/I H acts naturally [by conjugation] on I H ; we shall say that H is cuspidally central [relative to H → Π X ] if this action of H/I H on I H is trivial. Also, we shall use similar terminology to the terminology just introduced for H → Π X when Π X is replaced by Δ X , Π X×X , Δ X×X . (ii) Let H be a proﬁnite group; H 1 ⊆ H a closed subgroup. Then we shall refer to as an H 1 -inner automorphism of H an inner automorphism induced by conjugation by an element of H 1 . If H is also a proﬁnite group, then we shall refer to as an H 1 -outer homomorphism H → H an equivalence class of homo- morphisms H → H, where two such homomorphisms are considered equivalent if they diﬀer by composition by an H 1 -inner automorphism. If H is equipped with a homomorphism H → G k [cf., e.g., the various groups introduced above], and def H 1 = Ker(H → G k ), then we shall refer to an H 1 -inner automorphism (respec- tively, H 1 -outer homomorphism) as a geometrically inner automorphism (respec- tively, geometrically outer homomorphism). If H is equipped with a structure of extension of some other proﬁnite group H 0 by a ﬁnite product H 1 of copies of M X , or, more generally, a projective limit H 1 of such ﬁnite products, then we shall refer to an H 1 -inner automorphism (respectively, H 1 -outer homomorphism) as a cyclo- tomically inner automorphism (respectively, cyclotomically outer homomorphism). If H is equipped with a homomorphism to Π X , Δ X , Π X×X , or Δ X×X [cf. the situation of (i)], and H 1 is the kernel of this homomorphism, then we shall refer to an H 1 -inner automorphism (respectively, H 1 -outer homomorphism) as a cuspidally inner automorphism (respectively, cuspidally outer homomorphism). Next, let Π X ⊆ Π X be an open normal subgroup, corresponding to a ﬁnite étale Galois covering X → X. Set def Π Z = Π X ×X · Π X ⊆ Π X×X [where we regard Π X as a subgroup of Π X×X via the diagonal map]; write Z → X × X for the covering determined by Π Z . Thus, it is a tautology that the diagonal morphism ι : X → X × X lifts to a morphism ι : X → Z ABSOLUTE ANABELIAN CUSPIDALIZATIONS 11 which induces the inclusion Π X → Π Z on fundamental groups. If Z → X × X is a connected ﬁnite étale covering arising from an open subgroup of Π X×X , write: def U X×X = (X × X)\ι(X); def U Z = (U X×X ) × (X×X) Z Denote by Δ U X×X the maximal cuspidally [i.e., relative to the natural map to π 1 ((X × X) k )] pro-Σ † quotient of the maximal pro-Σ quotient of the tame funda- mental group of (U X×X ) k [where “tame” is with respect to the divisor ι(X) ⊆ X × X] and by Π U X×X the quotient π 1 (U X×X )/Ker(π 1 ((U X×X ) k ) Δ U X×X ); write Π U Z ⊆ Π U X×X for the open subgroup corresponding to the ﬁnite étale cover- ing U Z → U X×X . Proposition 1.6. (Characteristic Class of the Diagonal) (i) The pull-back morphism arising from the natural inclusion Π X → Π Z (⊆ Π X×X = Π X × G k Π X ) composed with the natural isomorphism of Proposition 1.2, (i), determines a homo- morphism ∼ † H d k +2 (Π Z , M X ⊗ M k ) → H d k +2 (Π X , M X ⊗ M k ) → Z hence [by Proposition 1.3, (ii)] a class diag ∈ H 2 (Π Z , M X ) η Z which is equal to the étale cohomology class associated to ι (X) ⊆ Z , or, alterna- tively, the [ﬁrst] Chern class of the line bundle O Z (ι (X)). (ii) Denote by L × diag [Z ] → Z the complement of the zero section in the geometric line bundle [i.e., G m -torsor] determined by O Z (ι (X)), by Δ L × [Z ] the maximal cuspidally pro-Σ † quotient of diag the maximal pro-Σ quotient of the tame fundamental group of (L × diag [Z ]) k [where “tame” is with respect to the divisor determined by the complement of the G m - 1 torsor L × diag [Z ] in the naturally associated P -bundle], and by Π L × [Z ] the quotient diag × π 1 (L × ). Then [in light of the isomorphism diag [Z ])/Ker(π 1 ((L diag [Z ]) k ) Δ L × diag [Z ] of Proposition 1.2, (ii)] we have a natural exact sequence 1 → M X → Π L × diag [Z ] → Π Z → 1 diag whose associated extension class is equal to the class η Z . (iii) The global section of O Z (ι (X)) over Z determined by the natural inclu- sion O Z → O Z (ι (X)) deﬁnes a morphism U Z → L × diag [Z ] 12 SHINICHI MOCHIZUKI over Z which induces a surjective homomorphism of groups over Π Z : Π U Z Π L × diag [Z ] Proof. Assertion (i) follows immediately from Propositions 1.1, 1.2, 1.3, together with well-known facts concerning Chern classes and associated cycles in étale co- homology [cf., e.g., [FK], Chapter II, Deﬁnition 1.2, Proposition 2.2]. Assertion (ii) follows from Proposition 1.1; [Mzk7], Deﬁnition 4.2, Lemmas 4.4, 4.5. Asser- tion (iii) follows from [Mzk8], Lemma 4.2, by considering ﬁbers over one of the two natural projections Π Z → Π X×X Π X . [Here, we note that although in [Mzk7], §4; [Mzk8], the base ﬁeld is assumed to be of characteristic zero, one ver- iﬁes immediately that the same arguments as those applied in loc. cit. yield the corresponding results in the ﬁnite ﬁeld case — so long as we restrict the coeﬃcients † .] of the cohomology modules in question to modules over Z Deﬁnition 1.7. (i) We shall refer to a covering Z → X × X as in the above discussion as the diagonal covering associated to the covering X → X. We shall refer to an extension of proﬁnite groups 1 → M X → D → Π Z → 1 diag whose associated extension class is the class η Z of Proposition 1.6, (i), as a fun- damental extension [of Π Z ]. In the following (ii) — (iv), we shall assume that 1 → M X → D → Π X×X → 1 is a fundamental extension. (ii) Let x, y ∈ X(k); write D x , D y ⊆ Π X for the associated decomposition groups [which are well-deﬁned up to conjugation by an element of Δ X — cf. Remark 1.7.1 below]. Now set: def D x = D| D x × Gk Π X ; def D x,y = D| D x × Gk D y Thus, D x (respectively, D x,y ) is an extension of Π X (respectively, G k ) by M X . Similarly, if D = i m i · x i , E = j n j · y j are divisors on X supported on points that are rational over k, then set: def D D = i m i · D x i ; def D D,E = m i · n j · D x i ,y j i,j [where the sums are to be understood as sums of extensions of Π X or G k by M X — i.e., the sums are induced by the additive structure of M X ]. Also, we shall write def C = −D| Π X [where we regard Π X as a subgroup of Π X×X via the diagonal map]. [Thus, C is an extension of Π X by M X whose extension class is the Chern class of the canonical bundle of X.] ABSOLUTE ANABELIAN CUSPIDALIZATIONS 13 (iii) Let S ⊆ X(k) be a ﬁnite subset. Then we shall write def D S = D x x∈S [where the product is to be understood as the ﬁber product over Π X ]. Thus, D S is an extension of Π X by a product of copies of M X indexed by elements of S. We shall refer to D S as a maximal abelian S-cuspidalization [of Π X at S]. Observe that if T ⊆ X(k) is a ﬁnite subset such that S ⊆ T , then we obtain a natural projection morphism D T → D S . (iv) We shall refer to a homomorphism Π U X×X → D over Π X×X as a fundamental section if, for some isomorphism of D with Π L × diag that induces the identity on Π X×X , M X , the resulting composite homomorphism Π U X×X → Π L × is the homomorphism of Proposition 1.6, (iii). diag Remark 1.7.1. Relative to the situation in Deﬁnition 1.7, (ii), conjugation by elements δ ∈ Δ X induces isomorphisms between the diﬀerent possible choices of “D x ”, all of which lie over the isomorphism between any of these choices and G k induced by the projection Π X G k . Moreover, by lifting (δ, 1) ∈ Δ X×X ⊆ Π X×X to an element δ D ∈ D, and conjugating by δ D , we obtain natural isomorphisms between the various resulting “D x ’s” which induce the identity on the quotient group D x Π X , as well as on the subgroup M X ⊆ D x . Note that this last property [i.e., of inducing the identity on Π X , M X ] holds precisely because we are working with δ ∈ Δ X ⊆ Π X , as opposed to an arbitrary “δ ∈ Π X ”. Remark 1.7.2. By Proposition 1.4, (ii), if E is any proﬁnite group extension of Π X (respectively, G k ; an open subgroup Π Z ⊆ Π X×X that surjects onto G k ) by M X , then the group of cyclotomically outer automorphisms of the extension E [i.e., that induce the identity on Π X (respectively, G k ; Π Z ) and M X ] may be naturally ∧ identiﬁed with (k × ) . In particular, in the context of Deﬁnition 1.7, (iv), any two fundamental sections of D diﬀer, up to composition with a cyclotomically inner ∧ automorphism of D, by a “(k × ) -multiple”. Next, if k is nonarchimedean local, then set G † k = G k ; if k is ﬁnite, then write † ]. Also, we shall use G † k ⊆ G k for the maximal pro-Σ † subgroup of G k [so G † k ∼ = Z the notation def Π † (−) = Π (−) × G k G † k ⊆ Π (−) def [where “(−)” is any smooth, geometrically connected scheme over k, with arithmetic fundamental group Π (−) G k ]. 14 SHINICHI MOCHIZUKI Proposition 1.8. tions) Let (Basic Properties of Maximal Abelian Cuspidaliza- 1 → M X → D → Π X×X → 1 be a fundamental extension; φ : Π U X×X D a fundamental section; S ⊆ X(k) a ﬁnite subset. Then: (i) The proﬁnite groups Δ X×X , Δ X , as well as any proﬁnite group extension or Π † X by a [possibly empty] ﬁnite product of copies of M X is slim [cf. of Π † X×X §0]. In particular, the proﬁnite group D S † = D S × G k G † k is slim. def def (ii) For x ∈ X(k), write U x = X\{x}. Denote by Δ U x the maximal cuspidally [i.e., relative to the natural map to π 1 ((U x ) k )] pro-Σ † quotient of the maximal pro-Σ quotient of the tame fundamental group of (U x ) k [where “tame” is with respect to the complement of U x in X] and by Π U x the quotient π 1 (U x )/Ker(π 1 ((U x ) k ) Δ U x ). Then the inverse image via either of the natural projections Π U X×X Π X of the decomposition group D x ⊆ Π X is naturally isomorphic to Π U x . In particular, Δ U X×X , Π † U X×X are slim. (iii) For S ⊆ X(k) a ﬁnite subset, write: def U S = U x x∈S [where the product is to be understood as the ﬁber product over X]. Denote by Δ U S the maximal cuspidally [i.e., relative to the natural map to π 1 ((U S ) k )] pro-Σ † quotient of the maximal pro-Σ quotient of the tame fundamental group of (U S ) k [where “tame” is with respect to the complement of U S in X], and by Π U S the quotient π 1 (U S )/Ker(π 1 ((U S ) k ) Δ U S ). Then Δ U S , Π † U S are slim. Forming the product of the specializations of φ to the various D x × G k Π X ⊆ Π X×X yields homomorphisms Π U x → D S Π U S → x∈S [where the product is to be understood as the ﬁber product over Π X ]. Moreover, the def composite morphism Π U S → D S is surjective; the resulting quotient of Δ U S = Ker(Π U S G k ) is the maximal cuspidally central quotient of Δ U S , relative to the surjection Δ U S Δ X . def (iv) The quotient of Δ U X×X = Ker(Π U X×X G k ) determined by φ : Π U X×X D is the maximal cuspidally central quotient of Δ U X×X , relative to the sur- jection Δ U X×X Δ X×X . Proof. Assertion (i) follows immediately from the slimness of Π † X , Δ X [cf., e.g., [Mzk5], Theorem 1.1.1, (ii); the proofs of [Mzk5], Lemmas 1.3.1, 1.3.10], together with the [easily veriﬁed] fact that G † k acts faithfully on M X via the cyclotomic char- acter. Next, we consider assertion (ii). The portion of assertion (ii) concerning Π U x ABSOLUTE ANABELIAN CUSPIDALIZATIONS 15 follows immediately from the existence of the “homotopy exact sequence associated to a family of curves” [cf., e.g., [Stix], Proposition 2.3]. The slimness assertion then follows from assertion (i) [applied to Π † X ] and the slimness of Δ U x [cf. the proofs of [Mzk5], Lemmas 1.3.1, 1.3.10]. As for assertion (iii), the slimness of Δ U S , Π † U S follows via the arguments given in the proofs of [Mzk5], Lemmas 1.3.1, 1.3.10. The existence of homomorphisms Π U S → x∈S Π U x → D S as asserted is immediate from the deﬁnitions, assertion (ii). For x ∈ S, write D x [U S ] ⊆ Π U S for the decomposition group of x; I x [U S ] ⊆ D x [U S ] for the inertia subgroup. Now it is immediate from the deﬁnitions that I x [U S ] maps isomorphically onto the copy M X in D S corresponding to the point x. This implies the desired surjectivity. Since, moreover, it is immediate from the deﬁnitions that the cuspidal subgroup of any cuspidally central quotient of Δ U S is generated by the image of the I x [U S ], as x ranges over the elements of S, the ﬁnal assertion concerning the maximal cuspidally central quotient of Δ U S follows immediately. Assertion (iv) follows by a similar argument to the argument applied to the ﬁnal portion of assertion (iii). Next, let Z → X × X (respectively, Z → X × X; Z ∗ → X × X) be the diagonal covering associated to a covering X → X (respectively, X → X; X ∗ → X) arising from an open subgroup of Π X ; denote by ι : X → Z (respectively, ι : X → Z ; ι ∗ : X → Z ∗ ) the tautological lifting of the diagonal embedding ι : X → X × X and by k (respectively, k ; k ∗ ) the extension of k determined by X (respectively, X ; X ∗ ). Assume, moreover, that the covering X → X factors as follows: X → X → X ∗ → X Thus, we obtain a factorization Z → Z → Z ∗ → X × X. Let 1 → M X → D → Π Z → 1 be a fundamental extension of Π Z . Write 1 → M X → D X ×X → Π X ×X → 1 for the pull-back of the extension D via the inclusion Π X ×X ⊆ Π Z . Now if we think of Π X×X or Π X ×X as only being deﬁned up to Δ X × {1}-inner automorphisms, then it makes sense, for δ ∈ Δ X /Δ X to speak of the pull-back of the extension D X ×X via δ × 1: 1 → M X → (δ × 1) ∗ D X ×X → Π X ×X → 1 In particular, we may form the ﬁber product over Π X ×X : def S X /X ∗ (D ) X ×X = δ∈Δ X ∗ /Δ X (δ × 1) ∗ D X ×X 16 SHINICHI MOCHIZUKI Thus, S X /X ∗ (D ) X ×X is an extension of Π X ×X by a product of copies of M X indexed by Δ X ∗ /Δ X ; S X /X ∗ (D ) X ×X admits a tautological Δ X × {1}- outer [more precisely: a (Δ X × {1}) × Π X ×X S X /X ∗ (D ) X ×X -outer] action by the ﬁnite group Δ X ∗ /Δ X ∼ = (Δ X ∗ /Δ X ) × {1}. Moreover, the natural outer ∼ action of Gal(X /X) = Gal((X × X )/Z ) ∼ = Π X /Π X on Π X ×X [arising from the diagonal embedding Π X → Π Z ] clearly lifts to an outer action of Gal(X /X) on S X /X ∗ (D ) X ×X , which is compatible, relative to the natural ac- tion of Gal(X /X) on Δ X ∗ /Δ X by conjugation, with the Δ X × {1}-outer action of Δ X ∗ /Δ X on S X /X ∗ (D ) X ×X . Thus, in summary, the natural isomorphism (Δ X ∗ /Δ X ) × {1} Gal(X /X) ∼ = Gal((X × X )/Z ∗ ) determines a homomorphism Gal((X × X )/Z ∗ ) → Out(S X /X ∗ (D ) X ×X ) via which we may pull-back the extension “1 → (−) → Aut(−) → Out(−) → 1” [cf. §0; Proposition 1.8, (i)] for S X /X ∗ (D ) X ×X to obtain an extension 1 → M X → S X /X ∗ (D ) → Π Z ∗ → 1 Δ X ∗ /Δ X in which Π Z ∗ is only determined up to Δ X × {1}-inner automorphisms. Note, moreover, that every cyclotomically outer automorphism of the extension D — ∧ i.e., an element of (k × ) [cf. Remark 1.7.2] — induces a cyclotomically outer automorphism of S X /X ∗ (D ). In particular, we have a natural cyclotomically outer ∧ action of (k × ) on S X /X ∗ (D ). Next, let us push-forward the extension S X /X ∗ (D ) just constructed via the natural surjection M X M X Δ X ∗ /Δ X Δ X ∗ /Δ X [which induces the identity morphism M X → M X between the various factors of the domain and codomain], so as to obtain an extension Tr X /X :X ∗ (D ) as follows: M X → Tr X /X :X ∗ (D ) → Π Z ∗ → 1 1 → Δ X ∗ /Δ X [in which Π Z ∗ is only determined up to Δ X × {1}-inner automorphisms]. Proposition 1.9. discussion above: (Symmetrizations and Traces) In the notation of the (i) The extension Tr X /X :X (D ) of Π Z by M X is a fundamental exten- sion of Π Z . (ii) There is a natural commutative diagram: 1 −→ M X −→ S X /X (D ) Δ X /Δ X 1 −→ Δ X /Δ X M X −→ S X /X (Tr X /X :X (D )) Π X×X −→ 1 −→ 1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ −→ id −→ Π X×X ABSOLUTE ANABELIAN CUSPIDALIZATIONS 17 [which is well-deﬁned up to Δ X × {1}-inner automorphisms — cf. Remark 1.9.1 below]. (iii) Relative to the commutative diagram of (ii), the natural cyclotomically ∧ ∧ outer action of (k × ) on S X /X (D ) lies over the composite of the map (k × ) → ∧ (k × ) given by raising to the [Δ X : Δ X ]-power with the natural cyclotomically ∧ outer action of (k × ) on S X /X (Tr X /X :X (D )). In particular, if N is a positive integer that divides [Δ X : Δ X ], then the natural cyclotomically outer action of ∧ an element of (k × ) on S X /X (D ) lies over the cyclotomically outer action of an ∧ element of {(k × ) } N on S X /X (Tr X /X :X (D )). Proof. that To verify assertion (i), observe that it is immediate from the deﬁnitions ι (X) × Z (X × X ) ⊆ X × X is equal to the Δ X /Δ X -orbit of ι (X)× Z (X × X ) ⊆ X × X . Now assertion (i) follows by translating this observation into the language of étale cohomology classes associated to subvarieties; assertions (ii), (iii) follow formally from assertion (i) and the deﬁnitions of the various objects involved. Remark 1.9.1. Relative to the commutative diagram of Proposition 1.9, (ii), note that, although S X /X (Tr X /X :X (D )) is, by deﬁnition, only well-deﬁned up to Δ X × {1}-inner automorphisms, the push-forward of S X /X (D ) by M X → M X Δ X /Δ X Δ X /Δ X is well-deﬁned up to Δ X × {1}-inner automorphisms. That is to say, the push- forward extension implicit in this commutative diagram furnishes a canonically more rigid version of the extension S X /X (Tr X /X :X (D )). Deﬁnition 1.10. (i) We shall refer to the extension S X /X ∗ (D ) [of Π Z ∗ ] constructed from the fundamental extension D as the [X /X ∗ -]symmetrization of D , or, alter- natively, as a symmetrized fundamental extension. We shall refer to the extension Tr X /X :X ∗ (D ) [of Π Z ∗ ] constructed from the fundamental extension D as the [X /X : X ∗ -]trace of D , or, alternatively, as a trace-symmetrized fundamental extension. (ii) If D is a fundamental extension of Π Z , then we shall refer to as a morphism of trace type any morphism S X /X (D ) → S X /X (D ) obtained by composing the morphism S X /X (D ) → S X /X (Tr X /X :X (D )) 18 SHINICHI MOCHIZUKI of Proposition 1.9, (ii), with a morphism S X /X (Tr X /X :X (D )) → S X /X (D ) arising [by the functoriality of the construction of “S X /X (−)”] from an isomorphism ∼ of [fundamental] extensions Tr X /X :X (D ) → D of Π Z by M X [which induces the identity on Π Z , M X ]. (iii) We shall refer to as a pro-symmetrized fundamental extension any com- patible system [indexed by the natural numbers] . . . S i . . . S j . . . Π X×X of morphisms of trace type [up to inner automorphisms of the appropriate type] be- tween symmetrized fundamental extensions, where S i is the X i /X-symmetrization of a fundamental extension of Π Z i ; Z i is the diagonal covering associated to an open normal subgroup Π X i ⊆ Π X ; the intersection of the Π X i is trivial. In this situation, we shall refer to the inverse limit proﬁnite group def lim S ∞ = ← − S i i as the limit of the pro-symmetrized fundamental extension {S i }; any proﬁnite group S ∞ arising in this fashion will be referred to as a pro-fundamental extension [of Π X×X ]. (iv) Let S ⊆ X(k) be a ﬁnite subset; S an X /X-symmetrization of a funda- mental extension D of Π Z . Then we shall write def S S = S D x × G Π X k x∈S [where the product is to be understood as the ﬁber product over Π X ]. Thus, S S is an extension of Π X by a product of copies of M X . Similarly, given a projective system {S i } as in (iii), we obtain a projective system {(S i ) S }, with inverse limit: (S ∞ ) S We shall refer to (S ∞ ) S as a maximal abelian S-pro-cuspidalization [of Π X at S]. Observe that if T ⊆ X(k) is a ﬁnite subset such that S ⊆ T , then we obtain a natural projection morphism (S ∞ ) T → (S ∞ ) S . Remark 1.10.1. Let D be as in Deﬁnition 1.7, (iii); S , {S i }, S ∞ as in Deﬁnition 1.10, (iii), (iv). Then observe that it follows from Proposition 1.8, (i), that the † [i.e., the result of applying “× G k G † k ” to “daggered versions” D † , (S ) † , S i † , and S ∞ D, S , S i , and S ∞ ] are slim. In particular, if S ⊆ X cl is any ﬁnite set of closed points of X, then we may form the objects D S † ; (S ) † S ; (S i ) † S ; (S ∞ ) † S ABSOLUTE ANABELIAN CUSPIDALIZATIONS 19 by passing to a Galois covering X k S → X [i.e., the result of base-changing X to some ﬁnite Galois extension k S of k] such that the closed points of X k S that lie over points of S are rational over k S ; forming the various objects in question over X k S [cf. Deﬁnition 1.7, (iii); Deﬁnition 1.10, (iv)]; and, ﬁnally, “descending to X” via the natural outer action of G k /G † k S on the various objects in question [cf. the exact sequence “1 → (−) → Aut(−) → Out(−) → 1” of §0; the slimness mentioned above]. Thus, in the remainder of this paper, we shall often speak of the various objects deﬁned in Deﬁnition 1.7, (iii); Deﬁnition 1.10, (iv), even when the points of the ﬁnite set S are not necessarily rational over k. Before proceeding, we note the following: Lemma 1.11. (Conjugacy Estimate) Let H ⊆ Δ X be a normal open subgroup; a ∈ Δ X /H an element not equal to the identity; N a Σ † -integer [cf. §0]. Then there exists a normal open subgroup H ⊆ Δ X contained in H such that for any normal open subgroup H ⊆ Δ X contained in H and any a ∈ Δ X /H that lifts a, the cardinality of the H-conjugacy class Conj(a , H ) ⊆ Δ X /H of a in Δ X /H is divisible by N . Proof. In the notation of the statement of Lemma 1.11, denote by Z(a , H ) ⊆ H the subgroup of elements δ ∈ H such that δ · a · δ −1 = a in Δ X /H . Then it is immediate that if a is the image of a in Δ X /H , then Z(a , H ) ⊆ Z(a , H ), so the cardinality of Conj(a , H ) ∼ = H/Z(a , H ) is divisible by the cardinality ∼ of Conj(a , H ) = H/Z(a , H ). Thus, it suﬃces to ﬁnd a normal open subgroup H ⊆ H such that for any a ∈ Δ X /H that lifts a, the cardinality of Conj(a , H ) is divisible by N . To this end, let us consider, for some prime number l ∈ Σ † , the maximal pro-l quotient H[l] of the abelianization H ab of H. Note that Δ X /H acts by conjugation on H ab , H[l]. Now I claim that there exists a [nonzero] h l ∈ H[l] such that a(h l ) = h l . Indeed, if this claim were false, then it would follow that a acts trivially on H[l]. But since a induces a nontrivial automorphism of the covering of X k determined by H, it follows that a induces a nontrivial automorphism of the l-power torsion points of the Jacobian of X k [since these points are Zariski dense in this Jacobian] — a contradiction. This completes the proof of the claim. Now let j ∈ H be an element that lifts the various h l obtained above for the [ﬁnite collection of] primes l that divide N ; let a X ∈ Δ X be an element that lifts a. Then observe that for some integer power M of N that is independent of the ab ⊗ (Z/M Z) is nonzero, for n ∈ Z choice of a X , the image of j n · a X · j −n · a −1 X in H Thus, if we take H equal to the inverse image of with nonzero image in Z/N · Z. M · H ab in H(⊆ Δ X ), we obtain that the intersection of the subgroup j Z ⊆ H with with nonzero Z(a , H ) [where a ∈ Δ X /H lifts a] does not contain j n , for n ∈ Z Z But this implies that the intersection (j ) Z(a , H ) ⊆ j N· Z , image in Z/N · Z. hence that [H : Z(a , H )] is divisible by N , as desired. 20 SHINICHI MOCHIZUKI Next, we consider the following fundamental extensions of Π Z , Π Z : D = Π L × def diag [Z ] D = Tr X /X :X (D ) def ; [cf. Proposition 1.6, (ii)]. Note that in this situation, it follows immediately from ∼ the deﬁnitions that we obtain a natural isomorphism D → Π L × [Z ] , which we shall diag use in the following discussion to identify D , Π L × [Z ] . Thus, we have fundamental diag sections: Π U Z D ; Π U Z D [cf. Proposition 1.6, (iii)]. In particular, by pulling back from Z to X × X , we obtain a surjection: Π U X ×X D X ×X Now if we apply the natural outer (Δ X /Δ X ) × {1}-action on Π U X ×X to this surjection, it follows from the deﬁnition of “S X /X (D )” that we obtain a natural homomorphism Π U X ×X S X /X (D ) X ×X which is easily veriﬁed [cf. Proposition 1.8, (ii), (iii)] to be surjective. Since, more- over, the construction of this surjective homomorphism is manifestly compatible with the outer actions of Gal(X /X) on both sides, we thus obtain a natural sur- jection: Π U X×X S X /X (D ) Now let us denote by D X ⊆ Π U X×X the decomposition group of the subvariety ι(X) ⊆ X × X. [Thus, D X is well-deﬁned up to conjugation; here, we assume that we have chosen a conjugate that maps to the image of the diagonal embedding Π X → Π X×X via the natural surjection Π U X×X Π X×X .] Observe that we have a natural exact sequence 1 → I X → D X → Π X → 1 [where I X — i.e., the inertia subgroup of D X — is deﬁned so as to make the sequence exact], together with a natural isomorphism I X ∼ = M X . Also, we shall def def write D X = D X Π U X ×X ; D X = D X Π U X ×X . Since the construction just carried out for double primed objects may also be carried out for single primed objects, we thus obtain the following: Proposition 1.12. (Symmetrized Fundamental Sections) In the notation of the discussion above: (i) There is a natural commutative diagram: D X ⏐ ⏐ ⊆ D X ⊆ id Π U X×X ⏐ ⏐ S X /X (D ) ⏐ ⏐ Π U X×X S X /X (D ) id ABSOLUTE ANABELIAN CUSPIDALIZATIONS 21 [where the vertical arrow on the right is the morphism in the diagram of Proposition 1.9, (ii)]. (ii) Denote by means of a subscript X the result of pulling back extensions of Π X×X , Π Z , Π X ×X to Π X [via the diagonal inclusion]. Then the projection [cf. the ﬁber product deﬁning S X /X (D )] to the factor labeled “Δ X /Δ X ” detemines a natural surjection ζ : S X /X (D ) X D X whose restriction to D X [i.e., relative to the arrows in the ﬁrst line of the com- ∼ mutative diagram of (i)] deﬁnes an isomorphism D X → D X . Moreover, the cuspidal subgroup of D X maps isomorphically onto the factor of M X in S X /X (D ) labeled “Δ X /Δ X ”. In particular, if we denote by S X /X (D ) = the quotient of S X /X (D ) by this factor of M X , then ζ determines a surjection ζ = : S X /X (D ) = X Π X ∼ whose restriction to the quotient D X Π X is equal to the identity Π X → Π X [up to geometric inner automorphisms]. Thus, we have a natural commutative diagram [well-deﬁned up to geometric inner automorphisms] ζ D X ⏐ ⏐ → S X /X (D ) X ⏐ ⏐ −→ Π X → S X /X (D ) = X −→ D X ⏐ ⏐ ζ = Π X in which the two horizontal composites are isomorphisms; the vertical arrows are surjections; both squares are cartesian. (iii) If we carry out the construction of (ii) for the single primed objects, then the commutative diagram of (i) induces a natural commutative diagram [well- deﬁned up to geometric inner automorphisms]: Π X ⏐ ⏐ Π X → S X /X (D ) = X ⏐ ⏐ → S X /X (D ) = X ζ = −→ Π X ⏐ ⏐ ζ = −→ Π X Moreover, there is a natural outer action of Gal(X /X) (respectively, Gal(X /X)) on the ﬁrst (respectively, second) line of this diagram; these outer actions are com- patible with one another. (iv) When considered up to cyclotomically inner automorphisms, the sections form a torsor over the group of ζ = (Δ X /Δ X )\(Δ X /Δ X ) ∧ ((k ) × ) 22 SHINICHI MOCHIZUKI [where the “\” denotes the set-theoretic complement]. The Gal(X /X)-equivariant form a torsor over the Gal(X /X)-invariant subgroup of this group. sections of ζ = Similar statements hold for the single primed objects. (v) The double and single primed torsors of equivariant sections of (iv) are related, via the right-hand square of the diagram of (iii), by a homomorphism ((k ) × ) ∧ (Δ X /Δ X ) \(Δ X /Δ X ) Gal(X /X) → ((k ) × ) ∧ Gal(X /X) (Δ X /Δ X ) \(Δ X /Δ X ) [where the superscripts denote the result of taking invariants with respect to the action of the superscripted group] that satisﬁes the following property: An element ξ of the domain maps to an element of the codomain whose component in the factor labeled a ∈ Δ X /Δ X is a product of elements of ∧ ((k ) × ) of the form N k /k (λ ) n . a ∧ Here, a ∈ (Δ X /Δ X )\(Δ X /Δ X ) maps to a in Δ X /Δ X ; λ ∈ ((k ) × ) is the component of ξ in the factor labeled a ; k a is an intermediate ﬁeld extension ∧ ∧ ∧ between k and k such that λ ∈ ((k a ) × ) ; N k a /k : ((k a ) × ) → ((k ) × ) is the norm map; n is the cardinality of the Δ X -conjugacy class of a in (Δ X /Δ X ). In particular, by Lemma 1.11 [where we take “H” to be Δ X , “H ” to be Δ X ], for a given Δ X , if, for a given positive integer N , Δ X is “suﬃciently small”, then an arbitrary Gal(X /X)-equivariant section of ζ = lies over the canonical section of ζ = given in (iii), up to the cyclotomically outer action of some N -th power of an element of the single primed version of the group exhibited in the display of (iv). Proof. All of these assertions follow immediately from the deﬁnitions [and, in the case of assertion (iv), Proposition 1.4, (ii)]. Deﬁnition 1.13. Let D be a fundamental extension of Π Z ; {S i } a pro- symmetrized fundamental extension, with limit S ∞ [cf. Deﬁnition 1.10, (iii)]. (i) We shall refer to as a symmetrized fundamental section a homomorphism Π U X×X S X /X (D ) obtained by composing the surjection Π U X×X S X /X (D ) of Proposition 1.12, ∼ (i), with the isomorphism S X /X (D ) → S X /X (D ) induced by an isomorphism ∼ D → D of fundamental extensions of Π Z by M X [which induces the identity on Π Z , M X ]. We shall refer to an inclusion D X → S X /X (D ) ABSOLUTE ANABELIAN CUSPIDALIZATIONS 23 obtained by restricting a symmetrized fundamental section to D X ⊆ Π U X×X [cf. Proposition 1.12, (i)] as a fundamental inclusion. (ii) We shall refer to a compatible system of symmetrized fundamental sections Π U X×X → S i as a pro-symmetrized fundamental section and to the resulting limit homomorphism Π U X×X → S ∞ as a pro-fundamental section. Similarly, we have a notion of “pro-fundamental inclusions”. Remark 1.13.1. Thus, by the above discussion, if we take the “S i ” to be the symmetrizations of the Π L × [Z ] as in Proposition 1.6, (ii), then we obtain natural diag pro-fundamental sections and pro-fundamental inclusions [cf. Proposition 1.12, (i), (ii), (iii)]. Proposition 1.14. (Maximal Cuspidally Abelian Quotients) Let {S i } be a pro-symmetrized fundamental extension, with limit S ∞ [cf. Deﬁnition 1.10, (iii)] and pro-fundamental section Π U X×X S ∞ [cf. Deﬁnition 1.13, (ii)]; S ⊆ X cl a ﬁnite set of closed points [cf. Remark 1.10.1]. Then: (i) The pro-fundamental section Π U X×X S ∞ determines a surjection Π U S (S ∞ ) S [cf. Proposition 1.8, (iii)]. The resulting quotient of Δ U S (respectively, Π U S ) is the maximal cuspidally abelian quotient Δ U S Δ c-ab U S (respectively, Π U S c-ab Π U S ) of Δ U S (respectively, Π U S ). (ii) The quotient of Δ U X×X (respectively, Π U X×X ) induced by the pro-funda- mental section Π U X×X S ∞ is the maximal cuspidally abelian quotient c-ab [which we shall denote by] Δ U X×X Δ c-ab U X×X (respectively, Π U X×X Π U X×X ) of Δ U X×X (respectively, Π U X×X ). Proof. Indeed, this follows as in the proof of Proposition 1.8, (iii), (iv), by observ- ing that the cuspidal subgroup of the maximal cuspidally abelian quotient of Δ U S (respectively, Δ U X×X ) is naturally isomorphic to the inverse limit of the cuspidal subgroups of the maximal cuspidally central quotients of the Δ U S × Δ X Δ X (⊆ Δ U S ) (respectively, Δ U X ×X ) [as Δ X ⊆ Δ X ranges over the open normal subgroups of Δ X ]. Proposition 1.15. (Automorphisms and Commensurators) Let {S i } be a pro-symmetrized fundamental extension, with limit S ∞ [cf. Deﬁnition 1.10, (iii)] and pro-fundamental inclusion D X → S ∞ [cf. Deﬁnition 1.13, (ii)]. Then: (i) Any automorphism α of the proﬁnite group Π c-ab U X×X which (a) is compatible with the natural surjection Π c-ab U X×X Π X×X and induces the identity on Π X×X ; 24 SHINICHI MOCHIZUKI (b) preserves the image of M X ∼ = I X ⊆ D X via the natural inclusion D X → Π c-ab U X×X is cuspidally inner. (ii) Π X (respectively, Δ X ) is commensurably terminal [cf. §0] in Π X×X (respectively, Δ X×X ). (iii) D X is commensurably terminal in S i , S ∞ ∼ = Π c-ab U X×X . Proof. First, we verify assertion (i). By Proposition 1.14, (ii), we have a natural ∼ isomorphism Π c-ab U X×X → S ∞ , so we may think of α as an automorphism of S ∞ . In light of (a); Proposition 1.8, (iii), it follows that α is compatible with the natural surjections S ∞ S i . Write α i for the automorphism of S i induced by α. By (a), (b), it follows that α i is an automorphism of the extension S i of Π X×X by a product of copies of M X which induces the identity on both Π X×X and the product of copies of M X [cf. the deﬁnition by a certain ﬁber product of the symmetrized fundamental extension S i ]. [Here, we note that the fact that α i induces the identity on each copy of M X follows by considering the non-torsion [cf. Propositions 1.2, (ii); 1.6, (i), (ii)] extension class determined by that copy of M X [which is preserved by α i !], together with the fact that α i induces the identity on the second cohomology groups of open subgroups of Δ X×X with coeﬃcients in M X .] Thus, up to cyclotomically inner ∧ automorphisms, α i arises from a collection of elements of (k i × ) , where k i is some ﬁnite Galois extension of k [cf. Proposition 1.4, (ii)], one corresponding to each copy of M X . Moreover, since these copies of M X are permuted by the action of Π X×X by conjugation, it follows that [up to cyclotomically inner automorphisms] ∧ ∧ ∧ α i arises from a single element of (k i × ) , which in fact belongs to (k × ) (⊆ (k i × ) ) [as one sees by considering the conjugation action via the “G k portion” of Π X×X ]. On the other hand, since the α i form a compatible system of automorphisms of the ∧ S i , it follows from Proposition 1.9, (iii), that this element of (k × ) must be equal to 1, as desired. Next, to verify assertion (ii), let us observe that it suﬃces to show that Δ X is commensurably terminal in Δ X×X . But this follows immediately from the fact that Δ X is slim [cf. Proposition 1.8, (i)]. Finally, we consider assertion (iii). Clearly, it suﬃces to show that D X is commensurably terminal in S i . By assertion (ii), to verify this commensurable terminality, it suﬃces to show that the [manifestly abelian] cuspidal subgroup H i ⊆ S i [i.e., relative to the natural surjection S i Π X×X ] satisﬁes the following property: Every h ∈ H i such that h δ − h ∈ D X , for all δ in some open subgroup J of D X , satisﬁes h ∈ D X . But this property follows immediately [cf. the deﬁnition by a certain ﬁber product of the symmetrized fundamental extension S i ] from the fact that, for J suﬃciently small, the J-module H i /(D X H i ) is isomorphic to a direct product of a ﬁnite number of copies of M X . The following result is the main result of the present §1: ABSOLUTE ANABELIAN CUSPIDALIZATIONS 25 Theorem 1.16. (Reconstruction of Maximal Cuspidally Abelian Quo- tients) Let X, Y be hyperbolic curves over a ﬁnite or nonarchimedean local ﬁeld; denote the base ﬁelds of X, Y by k X , k Y , respectively. Let Σ X (respectively, Σ Y ) be a set of prime numbers that contains at least one prime number that is invertible in k X (respectively, k Y ); write Δ X (respectively, Δ Y ) for the maximal cuspidally pro-Σ † X (respectively, pro-Σ † Y ) quotient of the maximal pro-Σ X (respectively, pro-Σ Y ) quotient of the tame fundamental group of X k X (re- spectively, Y k Y ) [where “tame” is with respect to the complement of X k X (respec- tively, Y k Y ) in its canonical compactiﬁcation], and Π X (respectively, Π Y ) for the corresponding quotient of the étale fundamental group of X (respectively, Y ). Let ∼ α : Π X → Π Y be an isomorphism of proﬁnite groups. Then: (i) We have Σ † X = Σ † Y ; write Σ † = Σ † X = Σ † Y . Moreover, k X is a ﬁnite ﬁeld if and only if k Y is; α preserves the decomposition groups of cusps; X is of type (g, r) [where g, r ≥ 0 are integers such that 2g − 2 + r > 0] if and only if Y is of type (g, r). Finally, if k X , k Y are nonarchimedean local, then their residue characteristics coincide. def (ii) α is compatible with the natural quotients Π X G k X , Π Y G k Y . (iii) Assume that X, Y are proper. Denote by Π U X×X Π c-ab U X×X , Π U Y ×Y c-ab Π U Y ×Y the maximal cuspidally [i.e., relative to the natural surjections Π U X×X Π X×X , Π U Y ×Y Π Y ×Y ] abelian quotients [cf. Proposition 1.14]. Then there is a commutative diagram [well-deﬁned up to cuspidally inner automorphisms] Π c-ab U X×X ⏐ ⏐ Π X×X α c-ab −→ Π c-ab U Y ×Y ⏐ ⏐ α×α −→ Π Y ×Y — where, the horizontal arrows are isomorphisms which are compatible with c-ab the natural inclusions D X → Π c-ab U X×X , D Y → Π U Y ×Y [cf. Proposition 1.12, (i)]; the vertical arrows are the natural surjections. Finally, the correspondence α → α c-ab is functorial [up to cuspidally inner automorphisms] with respect to α. Proof. First, we consider assertions (i), (ii). Note that k X is ﬁnite if and only if, for every open subgroup H ⊆ Π X , the quotient of the abelianization H ab by the closure of the torsion subgroup of H ab is topologically cyclic [cf. [Tama], Proposition 3.3, (ii)]; a similar statement holds for k Y , Π Y . Thus, k X is ﬁnite if and only if k Y is. Now suppose that k X , k Y are ﬁnite. Then assertion (ii) also follows from [Tama], 26 SHINICHI MOCHIZUKI Proposition 3.3, (ii). The fact that Σ † X = Σ † Y then follows from the following observation: The subset Σ † X ⊆ Primes is the subset on which the function Primes l → dim Q l ((Δ X ) ab ⊗ Q l ) attains its maximum value [cf. [Tama], Proposition 3.1]; a similar statement holds for Y . Now by considering the respective outer actions of G k X , G k Y on the max- imal pro-l quotients of Δ X , Δ Y , for some l ∈ Σ † , we obtain that α preserves the decomposition groups of cusps [hence that X is of type (g, r) if and only if Y is of type (g, r)], by [Mzk9], Corollary 2.7, (i). This completes the proof of assertions (i), (ii) in the ﬁnite ﬁeld case. Next, let us assume that k X , k Y are nonarchimedean local. Then the portion of assertion (i) concerning Σ X = Σ † X , Σ Y = Σ † Y follows by considering the cohomo- logical dimension of Π X , Π Y — cf., e.g., Proposition 1.3, (ii) [in the proper case]. def As for assertion (ii), if the cardinality of Σ = Σ † is ≥ 2, then assertion (ii) follows from the evident pro-Σ analogue of [Mzk5], Lemma 1.3.8; if the cardinality of Σ is 1, then assertion (ii) follows from Lemma 1.17, (c), (d) below. Now the portion of assertion (i) concerning the residue characteristics of k X , k Y follows from assertion (ii) and [Mzk5], Proposition 1.2.1, (i); the fact that α preserves the decomposition groups of cusps [hence that X is of type (g, r) if and only if Y is of type (g, r)] follows from [Mzk9], Corollary 2.7, (i). This completes the proof of assertions (i), (ii) in the nonarchimedean local ﬁeld case. Finally, we consider assertion (iii). It follows from the deﬁnitions that α induces ∼ → X, Z Y → Y are diagonal an isomorphism M X → M Y . If, moreover, Z X coverings corresponding to [connected] ﬁnite étale Galois coverings X → X, Y → Y that arise from open subgroups of Π X , Π Y that correspond via α, then α induces an isomorphism of group cohomology modules ∼ 2 , M X ) → H (Π Z , M Y ) H 2 (Π Z X Y , that preserves the extension classes associated to fundamental extensions of Π Z X Π Z Y [cf. Proposition 1.6, (i)]. In particular, if D (respectively, E ) is a fundamental (respectively, Π Z Y ), then α induces an isomorphism extension of Π Z X ∼ D → E ∼ ∼ → Π Z already induced which is compatible with the morphisms M X → M Y , Π Z X Y by α, and, moreover, uniquely determined, up to cyclotomically inner automor- ∧ × ∧ phisms, and the action of (k X ) (respectively, (k Y × ) ) [cf. Proposition 1.4, (ii)]. On the other hand, by allowing X , Y to vary, taking symmetrizations of the fundamen- tal extensions involved [which may be constructed entirely group-theoretically!], and making use of the vertical morphism in the center of the diagram of Proposition 1.9, (ii) [again an object which may be constructed entirely group-theoretically!], it follows from Proposition 1.9, (iii), that the indeterminacy of the isomorphism ∧ ∼ × ∧ D → E arising from the action of (k X ) , (k Y × ) “converges to the identity inde- ∼ terminacy” [i.e., by taking D → E to arise as just described from an isomorphism ABSOLUTE ANABELIAN CUSPIDALIZATIONS 27 ∼ of fundamental extensions D → E associated to [connected] ﬁnite étale coverings X → X , Y → Y [that arise from open subgroups of Π X , Π Y that correspond via α], where the open subgroups Π X ⊆ Π X , Π Y ⊆ Π Y are suﬃciently small]. Thus, in light of the manifest functoriality of the vertical morphism in the center of the diagram of Proposition 1.9, (ii) [the detailed explication of which, in terms of various commutative diagrams, is a routine task which we leave to the reader!], we obtain an isomorphism ∼ {S i } → {T j } of pro-symmetrized fundamental extensions [cf. Deﬁnition 1.10, (iii)] of Π X×X , Π Y ×Y , respectively, which arises from α and is completely determined up to cyclo- tomically inner automorphisms. Here, we pause to note that although in the con- struction of the symmetrization of a fundamental extension D (respectively, E ), one must, a priori, contend with a certain indeterminacy with respect to Δ X ×{1}- (respectively, Δ Y × {1}-)inner automorphisms [cf., e.g., Proposition 1.9, (ii)], in fact, by allowing X , Y to vary, this indeterminacy also “converges to the identity indeterminacy” [cf. Remark 1.9.1]. Thus, in summary, α induces an isomorphism [well-deﬁned up to cyclotomically [or, alternatively, cuspidally] inner automorphisms] ∼ S ∞ → T ∞ of pro-fundamental extensions of Π X×X , Π Y ×Y , respectively. Moreover, by apply- ing the fact that the left-hand square of the commutative diagram of Proposition ” 1.12, (ii), is cartesian, together with the fact that the “canonical section” of “ζ = that appears in Proposition 1.12, (iii), is completely determined [cf. Proposition 1.12, (v); Lemma 1.11] by the condition that it lie under an arbitrary “equivariant section” [cf. Proposition 1.12, (iv)] of the “ζ = ” associated to coverings “X → X ” arising from arbitrarily small open subgroups Π X ⊆ Π X , it follows that the isomor- ∼ phism S ∞ → T ∞ just obtained is compatible with the pro-fundamental inclusions D X → S ∞ , D Y → T ∞ . In particular, by Proposition 1.14, (ii) [cf. also Proposition 1.12, (i)], we conclude that α induces an isomorphism [well-deﬁned up to cuspidally inner automorphisms] ∼ c-ab (S ∞ ∼ = ) Π c-ab U X×X → Π U Y ×Y ( ∼ = T ∞ ) c-ab which is compatible with the natural inclusions D X → Π c-ab U X×X , D Y → Π U Y ×Y . Finally, the functoriality of this isomorphism follows from the naturality of its construction. Remark 1.16.1. It follows immediately from the naturality of the constructions used in the proof of Theorem 1.16, (iii), that when “α” arises from an isomorphism ∼ of schemes X → Y , the resulting α c-ab of Theorem 1.16, (iii), coincides with the morphism induced on fundamental groups by the resulting isomorphism of schemes ∼ U X×X → U Y ×Y . 28 SHINICHI MOCHIZUKI Lemma 1.17. (Normal Subgroups of the Absolute Galois Group of a Nonarchimedean Local Field) Let k be a nonarchimedean local ﬁeld of residue characteristic p; write G k for the absolute Galois group of k. Also, let us write I ⊆ G k for the inertia subgroup of G k and W ⊆ I for the wild inertia subgroup. [Here, we recall that W is the unique Sylow pro-p subgroup of I.] Let H ⊆ G k be a closed subgroup that satisﬁes [at least] one of the following four conditions: (a) H is a ﬁnite group. (b) H commutes with W . (c) H is a pro-prime-to-p group [i.e., the order of every ﬁnite quotient group of H is prime to p] that is normal in G k . (d) H is a topologically ﬁnitely generated pro-p group that is normal in G k . Then H = {1}. Proof. Indeed, suppose that H satisﬁes condition (a). Then the fact that H = {1} follows from [NSW], Corollary 12.1.3, Theorem 12.1.7. Now suppose that H satisﬁes condition (b). Then by the well-known functorial isomorphism [arising from local class ﬁeld theory] between the additive group underlying a ﬁnite ﬁeld extension of k that corresponds to an open subgroup J ⊆ G k and the tensor product with Q p of the image of W J in the abelianization J ab , it follows immediately that the conjugation action of H on W is nontrivial, whenever H is nontrivial. Thus we conclude again that H = {1}. Next, suppose that H satisﬁes condition (c). Then since H, W are both normal in G k , it follows [by considering commutators of elements of H with elements of W ] that arbitrary elements of H commute with arbitrary elements of W . In particular, H satisﬁes condition (b), so we conclude yet again that H = {1}. Finally, we assume that H is nontrivial and satisﬁes condition (d). Then I claim that H has trivial image Im(H) in G k /W . Indeed, since I/W , Im(H) are normal in G k /W , and, moreover, I/W is pro-prime-to-p, it follows that these two groups commute. On the other hand, since, as is well-known, G k /I acts faithfully [by conjugation, via the cyclotomic character] on I/W , it thus follows that Im(H) is trivial, as asserted. Thus, H ⊆ W . Since [as in well-known — cf., e.g., the proof of [Mzk4], Lemma 15.6] W is a free pro-p group of inﬁnite rank, we thus conclude that there exists an open subgroup U ⊆ W [so U is also a free pro-p group of inﬁnite rank] containing H such that the natural map H ab ⊗ F p → U ab ⊗ F p is injective, but not surjective. Then it follows immediately from the well-known theory of free pro-p groups that there exists a set of free topological generators {ξ i } i∈I [so the index set I is inﬁnite] of U such that for some ﬁnite subset J ⊆ I, ABSOLUTE ANABELIAN CUSPIDALIZATIONS 29 the elements {ξ j } j∈J lie in and topologically generate H. On the other hand, since H is normal in U , it follows from the well-known structure of free pro-p groups that we obtain a contradiction. This completes the proof of Lemma 1.17. Remark 1.17.1. The author would like to thank A. Tamagawa for informing him of the content of Lemma 1.17. Deﬁnition 1.18. In the situation of Theorem 1.16, (i), (ii), suppose further that def Σ X = Σ Y ; write Σ = Σ X = Σ Y . (i) If, for every ﬁnite étale covering X → X of X arising from an open subgroup Π X ⊆ Π X , it holds that the map from (X ) cl+ [cf. §0] to conjugacy classes of closed subgroups of Π X given by assigning to a closed point its associated decomposition group is injective, then we shall say that X is Σ-separated. (ii) If the map induced by α on closed subgroups of Π X , Π Y induces a bijection between the decomposition groups of the points of X cl+ , Y cl+ , then we shall say that α is quasi-point-theoretic. If α is quasi-point-theoretic, and, moreover, X, Y are Σ-separated — in which case α induces bijections ∼ X cl → Y cl ; ∼ X cl+ → Y cl+ — then we shall say that α is point-theoretic. (iii) Suppose further that we are in the ﬁnite ﬁeld case. Then we shall say ∼ that α is Frobenius-preserving if the isomorphism G k X → G k Y induced by α [cf. Theorem 1.16, (ii)] maps the Frobenius element of G k X to the Frobenius element of G k Y . Remark 1.18.1. In the ﬁnite ﬁeld case, when Σ † = Primes † , the Frobenius element of G k X may be characterized as in [Tama], Proposition 3.4, (i), (ii); a similar statement holds for the Frobenius element of G k Y . [Moreover, in the proper case, the Frobenius element of G k X may be characterized as the element of G k X that acts on M X via multiplication by the cardinality of k X , i.e., the cardinality of H 1 (G k X , M X ) plus 1.] Thus, when Σ † = Primes † , any α as in Theorem 1.16, (i), (ii), is automatically Frobenius-preserving. Remark 1.18.2. Let us suppose that we are in the situation of Deﬁnition 1.18, and that the base ﬁelds k X , k Y are ﬁnite. Let us refer to as a quasi-section [of Π X G k X ] any closed subgroup D ⊆ Π X [i.e., such as a decomposition group of a point ∈ X cl ] that maps isomorphically onto an open subgroup of G k X . Let us refer to a quasi-section of Π X G k X as a subdecomposition group if it is contained in some decomposition group of a point ∈ X cl . (i) Since X is not necessarily Σ-separated, it is not necessarily the case that decomposition groups of points ∈ X cl are commensurably terminal in Π X [cf. Propo- sition 2.6, (ii), below]. On the other hand, if D ⊆ Π X is a quasi-section, and we 30 SHINICHI MOCHIZUKI def write E = C Π X (D) ⊆ Π X for the commensurator of D in Π X [cf. §0], then one ver- iﬁes immediately E is also a quasi-section. [Indeed, by considering the projection Π X G k X , it follows immediately that every element of E centralizes some open subgroup D ⊆ D; on the other hand, by considering the well-known properties of the action of open subgroups of G k on abelianizations of open subgroups of Δ X [i.e., more precisely, the “Riemann hypothesis for abelian varieties over ﬁnite ﬁelds” — cf., e.g., [Mumf], p. 206], it follows that every centralizer of D in Δ X is trivial, i.e., that E Δ X = {1}.] (ii) It is immediate that any maximal subdecomposition group of Π X is, in fact, a decomposition group of some point ∈ X cl . On the other hand, since X is not necessarily Σ-separated, it is not clear whether or not every decomposition group of a point ∈ X cl is necessarily a maximal subdecomposition group. If X, Y are Σ-separated, then the arguments of [Tama], Corollary 2.10, Proposition 3.8, yield a “group-theoretic” characterization of the subdecomposition groups [hence also of the maximal subdecomposition groups, i.e., the decomposition groups of points ∈ X cl ] of Π X , Π Y in terms of the actions of the Frobenius elements. That is to say, if X, Y are Σ-separated, then any Frobenius-preserving isomorphism α is [quasi-]point- theoretic. (iii) Nevertheless, as was pointed out to the author by A. Tamagawa, even if X, Y are not necessarily Σ-separated, it is still possible to conclude, essentially from the arguments of [Tama], Corollary 2.10, Proposition 3.8, that: Any Frobenius-preserving isomorphism α is quasi-point-theoretic. Indeed, it suﬃces to give a “group-theoretic” characterization of the quasi-sections D ⊆ Π X which are decomposition groups of points ∈ X cl . We may assume [for def simplicity] without loss of generality that X, Y are proper. Write E = C Π X (D); k D , k E for the ﬁnite extension ﬁelds of k X determined by D, E. Let H ⊆ Δ X be a characteristic open subgroup; denote by Y → X the covering determined by the open subgroup E · H ⊆ Π X . Then it follows immediately from the deﬁnition of a “decomposition group” that it suﬃces to give a “group-theoretic” criterion for the condition that Y (k D ) contain a point whose ﬁeld of deﬁnition [which is, a priori, some subextension in k D of k E ] is equal to k D . In [Tama], the Lefschetz trace formula is applied to compute the cardinality of Y (k D ). On the other hand, if we use the superscript “ﬂd-def” to denote the subset of points whose ﬁeld of deﬁnition is equal to the ﬁeld given in parentheses, and “| − |” to denote the cardinality of a ﬁnite set, then for any subextension k ⊆ k D of k E , we have |Y (k )| = |Y (k ) ﬂd-def | k [where k ⊆ k ranges over the subextensions of k E ]. In particular, by applying induction on [k : k E ], one concludes immediately from the above formula that |Y (k ) ﬂd-def | may be computed from |Y (k )| for subextensions k ⊆ k of k E [while |Y (k )| may be computed, as in [Tama], from the Lefschetz trace formula]. This ABSOLUTE ANABELIAN CUSPIDALIZATIONS 31 yields the desired “group-theoretic” characterization of the decomposition groups of Π X . Remark 1.18.3. Note that in the ﬁnite ﬁeld case, if α as in Theorem 1.16, (i), (ii), is Frobenius-preserving, then the cardinalities, hence also the characteristics, of k X , k Y coincide. Indeed, this follows immediately by reducing to the proper case via Theorem 1.16, (i), and considering the actions of G k X , G k Y [cf. Theorem 1.16, ∼ (ii)] on M X , M Y [which are compatible relative to the isomorphism M X → M Y induced by α]. Now we return to the notation of the discussion preceding Theorem 1.16. Ob- serve that the automorphism τ : X × X → X × X given by switching the two factors induces an outer automorphism of Π U X×X . More- over, by choosing the basepoints used to form the various fundamental groups in- volved in an appropriate fashion, it follows that there exists an automorphism Π τ : Π U X×X → Π U X×X among those automorphisms induced by τ [i.e., all of which are related to one another by composition with an inner automorphism] which induces the automor- phism on Π X×X = Π X × G k Π X given by switching the two factors; preserves the subgroup D X ⊆ Π U X×X ; and preserves and induces the identity automorphism on the subgroup I X ⊆ D X (⊆ Π U X×X ). Note that by the slimness of Proposition 1.8, (i), together with the well-known commensurable terminality of D X ⊆ Π U X×X in Π U X×X [cf., e.g., [the proof of] [Mzk5], Lemma 1.3.12], it follows that, at least when Σ = Primes, these three conditions [are more than suﬃcient to] uniquely determine Π τ , up to composition with an inner automorphism arising from I X ; one then obtains a natural Π τ for arbitrary Σ [well-deﬁned up to composition with an inner automorphism arising from I X ] by taking the automorphism induced on the appropriate quotients by “Π τ in the case Σ = Primes”. Proposition 1.19. (Switching the Two Factors) The automorphism c-ab : Π c-ab Π c-ab τ U X×X → Π U X×X induced by Π τ is the unique automorphism of the proﬁnite group Π c-ab U X×X , up to composition with a cuspidally inner automorphism, that satisﬁes the following two conditions: (a) it preserves the quotient Π c-ab U X×X Π X×X and induces on this quotient the automorphism on Π X×X = Π X × G k Π X given by switching the two factors; (b) it preserves the image of I X ⊆ D X → Π c-ab U X×X . Proof. This follows immediately from Proposition 1.15, (i). 32 SHINICHI MOCHIZUKI Section 2: Points and Functions We maintain the notation of §1 [i.e., the discussion preceding Theorem 1.16]. If x ∈ X cl , then we shall denote by D x ⊆ Π X the decomposition group of x [well-deﬁned up to conjugation in Π X ]. If x ∈ X(k), then D x determines a section s x : G k → Π X [which is well-deﬁned as a geometrically outer homomorphism]. Next, let S ⊆ X cl be a ﬁnite set. If n is a Σ † -integer [cf. §0], then the Kummer exact sequence 1 → μ n → G m → G m → 1 [where G m → G m is the n-th power map; μ n is deﬁned so as to make the sequence exact] on the étale site of X determines a homomorphism Pic(X) → H 2 (Δ X , μ n ) [where Pic(X) is the Picard group of X]. Now there is a unique isomorphism ∼ μ n → M X /n · M X such that the homomorphism Pic(X) → H 2 (Δ X , μ n ) sends line bundles of degree 1 to the element determined by 1 ∈ Z/nZ via the composite of the induced iso- ∼ morphism H 2 (Δ X , μ n ) → H 2 (Δ X , M X /n · M X ) with the tautological isomorphism ∼ H 2 (Δ X , M X /n·M X ) → Z/nZ [cf. Proposition 1.2, (i)]. In the following discussion, we shall identify μ n with M X /n · M X via this isomorphism. If we consider the Kummer exact sequence on the étale site of U S ⊆ X [and pass to the inverse limit with respect to n], then we obtain a natural homomorphism × Γ(U S , O U ) → H 1 (Π U S , M X ) S [where we note that here, it suﬃces to consider the group cohomology of Π U S [i.e., as opposed to the étale cohomology of U S ], since the extraction of n-th roots of an × ) yields ﬁnite étale coverings of U S that correspond to open element of Γ(U S , O U S × subgroups of Π U S ] which is injective [since the abelian topological group Γ(U S , O U ) S † is clearly topologically ﬁnitely generated and free of p -torsion, hence injects into its prime-to-p † completion] whenever Σ † = Primes † . In particular, by allowing S to vary, we obtain a natural homomorphism × 1 K X → lim −→ H (Π U S , M X ) S [where K X is the function ﬁeld of X; the direct limit is over all ﬁnite subsets S of X cl ] which is injective whenever Σ † = Primes † . Proposition 2.1. subset, write (Kummer Classes of Functions) If S ⊆ X cl is a ﬁnite c-cn Δ U S Δ c-ab U S Δ U S ABSOLUTE ANABELIAN CUSPIDALIZATIONS 33 for the maximal cuspidally abelian and maximal cuspidally central quo- tients, respectively, and c-cn Π U S Π c-ab U S Π U S for the corresponding quotients of Π U S . If x ∈ X cl , then let us write D x [U S ] ⊆ Π U S for the decomposition group of x in Π U S [which is well-deﬁned up to conjugation in Π U S ] and I x [U S ] ⊆ D x [U S ] for the inertia subgroup. [Thus, when x ∈ S, we def obtain [cf. Proposition 1.6, (ii), (iii)] a natural isomorphism of M X with I x [U S ] = D x [U S ] Δ U S .] (i) The natural surjections induce isomorphisms as follows: ∼ ∼ 1 c-ab 1 H 1 (Π c-cn U S , M X ) → H (Π U S , M X ) → H (Π U S , M X ) In particular, we obtain natural homomorphisms as follows: ∼ ∼ × 1 c-ab 1 ) → H 1 (Π c-cn Γ(U S , O U U S , M X ) → H (Π U S , M X ) → H (Π U S , M X ) S ∼ ∼ × 1 c-cn 1 c-ab 1 → lim K X −→ H (Π U S , M X ) → lim −→ H (Π U S , M X ) → lim −→ H (Π U S , M X ) S S S These natural homomorphisms are injective whenever Σ † = Primes † . (ii) Suppose that S ⊆ X(k) is a ﬁnite subset. Then restricting cohomology classes of Π U S to the various I x [U S ], for x ∈ S, yields a natural exact sequence ∧ 1 → (k × ) → H 1 (Π U S , M X ) → † Z x∈S † ]. Moreover, the image [via the [where we identify Hom Z † (I x [U S ], M X ) with Z ∧ × natural homomorphism given in (i)] of Γ(U S , O U ) in H 1 (Π U S , M X )/(k × ) is equal S ∧ to the inverse image in H 1 (Π U S , M X )/(k × ) of the submodule of x∈S † Z ⊆ Z x∈S determined by the principal divisors [with support in S]. A similar statement c-cn holds when “Π U S ” is replaced by “Π c-ab U S ” or “Π U S ”. × (iii) If f ∈ Γ(U S , O U ), write S ∈ H 1 (Π c-cn κ c-cn f U S , M X ); κ c-ab ∈ H 1 (Π c-ab f U S , M X ); κ f ∈ H 1 (Π U S , M X ) for the associated Kummer classes. If x ∈ X cl \S, then D x [U S ] maps, via the natural surjection Π U S G k , isomorphically onto the open subgroup G k(x) ⊆ G k 34 SHINICHI MOCHIZUKI [where k(x) is the residue ﬁeld of X at x]. Moreover, the images of the pulled back classes ∼ | D x [U S ] = κ c-ab | D x [U S ] = κ f | D x [U S ] ∈ H 1 (D x [U S ], M X ) → H 1 (G k(x) , M X ) κ c-cn f f ∼ ∧ → (k(x) × ) ∧ ∧ in (k(x) × ) are equal to the image in (k(x) × ) of the value of f at x. Proof. Assertion (i) follows immediately from the deﬁnitions. The exact sequence of assertion (ii) follows immediately from Proposition 1.4, (ii). The characterization × ) is immediate from the deﬁnitions and the exact sequence of the image of Γ(U S , O U S of assertion (ii). Assertion (iii) follows immediately from the deﬁnitions and the functoriality of the Kummer class. Remark 2.1.1. If, in the situation of Proposition 2.1, (iii), we think of the extension of Π c-cn of Π X as being given by the extension D S [cf. Proposition 1.8, U S (iii)], where D is a fundamental extension of Π X×X that appears as a quotient of ∧ Π U X×X [hence is “rigid” with respect to the action of (k × ) — cf. Proposition 1.9, (iii); the proof of Theorem 1.16, (iii)], then it follows that the image of D x [U S ] in Π c-cn U S may be thought of as the image of D x [U S ] in D S . If, moreover, we assume, for simplicity, that x ∈ X(k), S ⊆ X(k), then this image of D x [U S ] in D S amounts to a section of D S Π X G k lying over the section s x of Π X G k . Since D S is deﬁned as a certain ﬁber product, this section is equivalent to a collection of sections [regarded as cyclotomically outer homomorphisms] γ y,x : G k → D y,x [where y ranges over the points of S]. [Here, we note that it is immediate from the deﬁnitions that, as the notation suggests, γ y,x depends only on x, y — i.e., that γ y,x is independent of the choice of S.] That is to say, from this point of view, Proposition 2.1, (iii), may be regarded as stating that: ∧ ∧ × ) The image in (k × ) = (k(x) × ) of the value of a function ∈ Γ(U S , O U S at x ∈ X(k) may be computed from its Kummer class, as soon as one knows the sections γ y,x : G k → D y,x , for y ∈ S. Also, before proceeding, we note that an arbitrary section of D y,x G k diﬀers [as a cyclotomically outer homomorphism] from γ y,x by the action of an element of ∼ ∧ H 1 (G k , M X ) → (k × ) . Thus, the datum of “γ y,x ” may be regarded as a trivializa- ∧ tion of a certain (k × ) -torsor. Remark 2.1.2. The ﬁnite ﬁeld portion of Proposition 2.1 may be regarded as the evident ﬁnite ﬁeld analogue of [a certain portion of] the theory of [Mzk8], §4. Also, we observe that the approach of “reconstructing the function ﬁeld of the curve via Kummer theory, as opposed to class ﬁeld theory [as was done in [Tama], [Uchi]]” ABSOLUTE ANABELIAN CUSPIDALIZATIONS 35 has the advantage of being applicable to nonarchimedean local ﬁelds, as well as to ﬁnite ﬁelds. Deﬁnition 2.2. For x, y ∈ X(k), we shall refer to the section [regarded as a cyclotomically outer homomorphism] γ y,x : G k → D y,x as the Green’s trivialization of D at (y, x). If D is a divisor on X supported in the subset of k-rational points X(k) ⊆ X cl , then multiplication of the various Green’s trivializations for the points in the support of D determines a section [regarded as a cyclotomically outer homomorphism] γ D,x : G k → D D,x which we shall refer to as the Green’s trivialization of D at (D, x). [Note that the deﬁnition of γ D,x generalizes immediately to the case where the divisor D, but not necessarily the points in its support, is rational over k — cf. Remark 1.10.1.] Remark 2.2.1. The terminology of Deﬁnition 2.2, is intended to suggest the similarity between the γ y,x of the present discussion and the “Green’s functions” that occur in the theory of bipermissible metrics — cf., e.g., [MB], §4.11.4. Remark 2.2.2. Note that the Green’s trivializations are symmetric with respect of Proposition 1.19. to the involution of D induced by the automorphism Π c-ab τ Indeed, relative to the natural projections Π U X×X Π c-ab U X×X D the Green’s trivialization at (y, x) is simply the section of D G k arising [by composition] from the section of Π U X×X G k determined by the decomposition group of the point (y, x) ∈ U X×X (k). Thus, the asserted symmetry of the Green’s is compatible with Π τ , together with trivializations follows from the fact that Π c-ab τ the evident fact that [by “transport of structure”] Π τ maps the decomposition group of (y, x) ∈ U X×X (k) isomorphically onto the decomposition group of (x, y) ∈ U X×X (k). If d ∈ Z, denote by J d the subscheme of the Picard scheme of X that parame- def trizes line bundles of degree d; write J = J 0 . Thus, J d is a torsor over J. Note that there is a natural morphism X → J 1 [given by assigning to a point of X the line bundle of degree 1 determined by the point]. Thus, the basepoint of X [already chosen in §1] determines a basepoint of J 1 . At the level of “geometrically pro-Σ” étale fundamental groups, this morphism induces a surjective homomorphism Π X Π J 1 36 SHINICHI MOCHIZUKI whose kernel is the kernel of the maximal abelian quotient Δ X Δ ab X . In partic- ular, for x ∈ X(k), the section s x determines a section t x : G k → Π J 1 . Note that applying the “change of structure group” given by the “multiplication by d map” on J to the J-torsor J 1 yields the J-torsor J d . [Indeed, this follows by considering the group structure of the Picard scheme.] Thus, we obtain a morphism J 1 → J d whose induced morphism on fundamental groups Π J 1 → Π J d determines an isomorphism of Π J d with the push-forward of the extension Π J 1 [i.e., ab ab of G k by Δ J 1 ∼ = Δ ab X ] via the homomorphism Δ X → Δ X given by multiplication by d. When d ≥ 1, the group structure on the Picard scheme also determines a morphism Π J 1 → Π J d [where the product is a ﬁber product over G k of d factors of Π J 1 ] which determines an isomorphism of Π J d with the push-forward ab of the ab extension constituted by the ﬁber product via the homomorphism Δ X → Δ X [i.e., from a product of d ab ab copies of Δ X to Δ X given by adding up the d components]. Moreover, one veriﬁes immediately that when d ≥ 1, these two constructions of “Π J d ” from Π J 1 yield groups that are naturally isomorphic. Thus, by applying the various homomorphisms induced on fundamental groups by the group structure of the Picard scheme, it follows that if D is any divisor of degree d on X whose support lies in the set of k-rational points X(k) ⊆ X cl , then D determines a section t D : G k → Π J d which may be constructed entirely group-theoretically from the “t x ”, where x ∈ X(k) ranges over the points in the support of D. In particular, if D is of degree 0, then the section t D : G k → Π J may be compared with the identity section of Π J to obtain a cohomology class: η D ∈ H 1 (G k , Δ ab X ) Now we have the following well-known result: Proposition 2.3. (Points and Galois Sections) Suppose that Σ = Primes. Then, in the notation of the above discussion: (i) The divisor D is principal if and only if η D = 0. (ii) The map x → D x from X cl to conjugacy classes of closed subgroups of Π X is injective, i.e., X is Primes-separated. Proof. First, we consider assertion (i). By well-known general nonsense [cf., e.g., [Naka], Claim (2.2); [NTs], Lemma (4.14); [Mzk4], the Remark preceding Deﬁnition 6.2], there is a natural isomorphism ∼ ∧ H 1 (k, Δ ab X ) → J(k) (⊇ J(k)) ABSOLUTE ANABELIAN CUSPIDALIZATIONS 37 [where the “∧” denotes the proﬁnite completion] which maps η D to the element of J(k) determined by D. [Here, we recall that this natural isomorphism arises by considering the long exact sequence obtained by applying the functors H ∗ (G k , −) to the short exact sequence of G k -modules 1 → J(k)[n] → J(k) → J(k) → 1 — where n is a positive integer; the morphism J(k) → J(k) is the “multiplication by n map”; J(k)[n] is deﬁned so as to make the sequence exact.] Thus, assertion (i) follows immediately. To prove assertion (ii), it suﬃces [by possibly base-changing to a ﬁnite exten- sion of k] to verify that two points x 1 , x 2 ∈ X(k) that induce Δ X -conjugate sections s x 1 , s x 2 are necessarily equal [cf. also [Tama], Corollary 2.10]. But this follows for- mally from assertion (i), by considering the divisor x 1 − x 2 [and the well-known fact that the natural morphism X → J 1 considered above is an embedding]. Remark 2.3.1. From the point of view of Deﬁnition 1.7, (ii), the reader may feel tempted to expect that [still under the assumption that Σ = Primes] D is principal if and only if the extension D D of Π X [by M X ] is trivial [i.e., determines the zero class in H 2 (Π X , M X )]. When k is nonarchimedean local, it is not diﬃcult to verify, using Proposition 2.3, (i), that this is indeed the case. On the other hand, when k is ﬁnite, although this condition for principality is easily veriﬁed to be necessary, it is not, however, suﬃcient, since it only involves the “prime-to-p † portion” of the point of J(k) determined by D. Deﬁnition 2.4. In the situation of Theorem 1.16, (iii), suppose further that def (Σ = ) Σ X = Σ Y , and that α is point-theoretic. Let S ⊆ X cl be a [not necessarily ∼ ﬁnite] subset that corresponds via the bijection X cl → Y cl induced by [the point- theoreticity of] α to a subset T ⊆ Y cl . (i) Write D (respectively, E) for the fundamental extension of Π X×X (respec- c-ab tively, Π Y ×Y ) that arises as the quotient of Π c-ab U X×X (respectively, Π U Y ×Y ) by the c-cn kernel of the maximal cuspidally central quotient Δ c-ab U X×X Δ U X×X (respectively, c-cn c-ab Δ c-ab induces an isomorphism: U Y ×Y Δ U Y ×Y ) [cf. Proposition 1.8, (iv)]. Thus, α ∼ α c-cn : D → E We shall say that α is (S, T )-locally Green-compatible if, for every pair of points (x 1 , x 2 ) ∈ X(k X ) × X(k X ) corresponding via the bijection induced by α to a pair of points (y 1 , y 2 ) ∈ Y (k Y ) × Y (k Y ), such that x 2 ∈ S, y 2 ∈ T , the isomorphism ∼ D x 1 ,x 2 → E y 1 ,y 2 [obtained by restricting α c-cn ] is compatible with the Green’s trivializations. We shall say that α is (S, T )-locally degree zero (respectively, (S, T )-locally principally) 38 SHINICHI MOCHIZUKI Green-compatible if, for every x ∈ X(k X ) S and every divisor of degree zero (respectively, principal divisor) D supported in X(k X ) ⊆ X cl corresponding via the bijection induced by α to a pair (y, E) of Y [so y ∈ Y (k Y ) T ], the isomorphism ∼ D D,x → E E,y is compatible with the Green’s trivializations. (ii) We shall say that α is totally (S, T )-locally Green-compatible (respectively, totally (S, T )-locally degree zero Green-compatible; totally (S, T )-locally principally Green-compatible) if, for all pairs of connected ﬁnite étale coverings X → X, Y → Y that arise from open subgroups of Π X , Π Y that correspond via α, the isomorphism ∼ Π X → Π Y induced by α is (S , T )-locally Green-compatible (respectively, (S , T )-locally de- gree zero Green-compatible; (S , T )-locally principally Green-compatible), where S ⊆ (X ) cl , T ⊆ (Y ) cl are the inverse images in X , Y of S, T , respectively. (iii) With respect to the terminology introduced in (i), (ii), when S = X cl , T = Y cl , then we shall replace the phrase “(S, T )-locally” by the phrase “globally”. Remark 2.4.1. In the situation of Deﬁnition 2.4, if X → X, Y → Y are con- nected ﬁnite étale coverings that arise from open subgroups of Π X , Π Y that corre- ∼ spond via α; D → E is the isomorphism of fundamental extensions of Π X×X , Π Y ×Y that arises from the isomorphism α c-ab of Theorem 1.16, (iii); and the points x 1 , x 2 (respectively, y 1 , y 2 ) are Δ X - (respectively, Δ Y -) conjugate, then it follows imme- diately from the compatibility of α c-ab with the natural inclusions D X → Π c-ab U X×X , ∼ D Y → Π c-ab U Y ×Y [cf. Theorem 1.16, (iii)] that the isomorphism D x 1 ,x 2 → E y 1 ,y 2 is automatically compatible with the Green’s trivializations. [Indeed, this follows from the easily veriﬁed fact that the Green’s trivializations in this case are, in essence, ” of Proposition 1.12.] specializations of conjugates of the “canonical sections of ζ = Unfortunately, however, the author is unable, at the time of writing, to see how to generalize the argument applied in the proof of Theorem 1.16, (iii), involving Lemma 1.11; Proposition 1.12, (v), so as to cover the case where the points x 1 , x 2 (respectively, y 1 , y 2 ) fail to be Δ X - (respectively, Δ Y -) conjugate. Remark 2.4.2. It is immediate that (S, T )-local Green-compatibility (respec- tively, (S, T )-local degree zero Green-compatibility) implies (S, T )-local degree zero Green-compatibility (respectively, (S, T )-local principal Green-compatibility), and that total (S, T )-local Green-compatibility (respectively, total (S, T )-local degree zero Green-compatibility) implies total (S, T )-local degree zero Green-compatibility (respectively, total (S, T )-local principal Green-compatibility). Theorem 2.5. (Reconstruction of Functions) In the situation of Theorem def 1.16, (iii), suppose further that (Σ = ) Σ X = Σ Y , and that α is point-theoretic. Then: ABSOLUTE ANABELIAN CUSPIDALIZATIONS 39 (i) Let S ⊆ X cl , T ⊆ Y cl be ﬁnite subsets that correspond via the bijection ∼ X cl → Y cl induced by α. Then α, α c-ab induce isomorphisms [well-deﬁned up to cuspidally inner automorphisms] ∼ c-ab Π c-ab U S → Π V T def [where V T = Y \T ] lying over α, which are functorial with respect to α and S, T , as well as with respect to passing to connected ﬁnite étale coverings of X, Y [that do not necesarily arise from open subgroups of Π X , Π Y !]. ∼ (ii) Suppose that Σ = Primes. Then the bijection X cl → Y cl induced by α in- duces a bijection between the groups of principal divisors on X, Y . This bijection, together with the isomorphisms of (i), induces a compatible isomorphism ∧ ∼ × × · (k X ) → K Y × · (k Y × ) K X ∧ between the push-forwards of the multiplicative groups associated to the function ∧ × × ∧ ﬁelds of X, Y , relative to the homomorphisms k X → (k X ) , k Y × → (k Y × ) . Proof. Assertion (i) follows immediately by “specializing to S, T ” the isomorphism of Theorem 1.16, (iii) [cf. also Proposition 1.14, (i), (ii); the deﬁnitions of the various objects involved]. [Here, we note that the functoriality asserted in assertion (i), which is somewhat stronger than the functoriality asserted in Theorem 1.16, (iii), follows from the deﬁnitions, together with the naturality of the constructions applied in the proof of Theorem 1.16, (iii) — cf., e.g., the diagram of Proposition 1.9, (ii).] Assertion (ii) follows immediately from assertion (i); Proposition 2.3, (i); Proposition 2.1, (i), (ii). ∼ c-ab of Theorem Remark 2.5.1. In fact, the crucial isomorphism Π c-ab U S → Π V T 2.5, (i), may also be constructed, in the ﬁnite ﬁeld case, via the techniques to be introduced in §3 [although we shall not discuss this approach in detail; cf., however, the proof of Theorem 3.10]. On the other hand, observe that unlike the techniques of §3, the techniques of §1 [in particular, the proof of Theorem 1.16, (iii), via Propositions 1.9, 1.12] apply to situations [e.g., the case of nonarchimedean local ﬁelds!] where the weight ﬁltration [cf. §3] does not admit a Galois-invariant splitting. Indeed, the techniques of §1, essentially only require that the Galois cohomology of the base ﬁeld admit a natural duality pairing. Moreover, even in the ∼ c-ab ﬁnite ﬁeld case, in light of the importance of this isomorphism Π c-ab U S → Π V T in the theory of the present paper, it is of interest to see that this isomorphism may be constructed via two fundamentally diﬀerent approaches. Finally, although the techniques of §3 are better suited to the reconstruction of the Green’s trivializations, they have the drawback that they depend essentially on the choice of a “basepoint” x ∗ ∈ X(k). Thus, it is of interest to know that this isomorphism may be constructed [i.e., via the techniques of §1] “cohomologically” [cf. Proposition 1.6, (i)] without making such a choice. 40 SHINICHI MOCHIZUKI Remark 2.5.2. In the case of nonarchimedean local ﬁelds, it is natural to ask, in the style of [Mzk8], §4, whether or not various “canonical integral structures” on the extensions D x,y [where x, y ∈ X(k)] of G k by M X are preserved by arbitrary isomorphisms of arithmetic fundamental groups. When x = y, such a canonical integral structure is determined by the Green’s trivialization; when x = y, such a canonical integral structure is determined by the integral structure [in the usual sense of scheme theory] on the canonical sheaf of the stable model of the curve [when the curve has stable reduction] — cf. [Mzk8], §4. Before proceeding, we note the following “analogue for Π c-ab U S ” of Proposition 1.15, (i): Proposition 2.6. (Automorphisms and Commensurators) Let Π c-ab U S be c-ab as in Proposition 2.1. For x ∈ S, write D x [U S ] → Π U S for the natural inclusion. Then: (i) Any automorphism α of the proﬁnite group Π c-ab U S which (a) is compatible with the natural surjection Π c-ab U S Π X and induces the identity on Π X ; (b) for each x ∈ S, preserves the image of M X ∼ = I x [U S ] ⊆ D x [U S ] via the c-ab natural inclusion D x [U S ] → Π U S is cuspidally inner. (ii) Suppose that X is Σ-separated. Then for x ∈ S, D x is commensurably terminal in Π X . (iii) Suppose that X is Σ-separated. Then the image of D x [U S ] → Π c-ab U S is c-ab commensurably terminal in Π U S . Proof. First, we observe that assertion (ii) follows formally from the deﬁnition of a “decomposition group” and “Σ-separated”. Thus, assertion (i) (respectively, (iii)) follows by an argument which is entirely similar to the argument that was used to prove assertion (i) (respectively, (iii)) of Proposition 1.15. Remark 2.6.1. In the situation of Deﬁnition 2.4, suppose that S, T are ﬁnite, and that α arises from an isomorphism ∼ Π U S → Π V T which is point-theoretic [or, equivalently, quasi-point-theoretic] — a condition that is automatically satisﬁed in the ﬁnite ﬁeld case whenever α is Frobenius-preserving [cf. ABSOLUTE ANABELIAN CUSPIDALIZATIONS 41 Remark 1.18.2]. Then observe that, [in light of our point-theoreticity assumption] it follows from Proposition 2.6, (i), that the resulting induced isomorphism ∼ c-ab Π c-ab U S → Π V T coincides [up to cuspidally inner automorphisms] with the isomorphism of Theorem 2.5, (i). Thus, in light of Remark 2.2.2, it follows formally from the deﬁnitions that α is totally (S, T )-locally Green-compatible. Corollary 2.7. (Point-theoretic Totally Locally Principally Green- compatible Isomorphisms) In the situation of Theorem 1.16, (iii), assume fur- def ther that (Σ = ) Σ X = Σ Y = Primes, and that α is point-theoretic and to- tally (S, T )-locally principally Green-compatible, for some nonempty sub- ∼ sets S ⊆ X cl , T ⊆ Y cl which correspond via the bijection X cl → Y cl induced by α. Then α arises from a uniquely determined commutative diagram of schemes ∼ X ⏐ ⏐ → X → ∼ Y ⏐ ⏐ Y in which the horizontal arrows are isomorphisms; the vertical arrows are the pro- ﬁnite étale coverings determined by the proﬁnite groups Π X , Π Y . Proof. Corollary 2.7 follows immediately — i.e., by “specializing functions to points” — from the deﬁnitions; Theorem 2.5, (ii); Proposition 2.1, (iii); Remark 2.1.1; and [Tama], Lemma 4.7. Here, we note that, in the present situation, the isomorphism ∧ × × ∧ ∼ · (k X ) → K Y × · (k Y × ) K X ∼ × of Theorem 2.5, (ii), necessarily induces an isomorphism K X → K Y × [cf. the as- sumption that Σ † = Primes † ]. Indeed, this is immediate in the ﬁnite ﬁeld case. In the nonarchimedean local ﬁeld case, it follows via the arguments applied in the proof of [Mzk8], Theorem 4.10: That is to say, we assume for simplicity that S ⊆ X(k X ); × then if f ∈ K X , and x ∈ S is a point that does not lie in the divisor of zeroes and poles of f , then let us observe that the subset × × ⊆ f · (k X ) f · k X ∧ may be characterized as the subset of elements whose values [cf. Proposition 2.1, × × ∧ ⊆ (k X ) . Note that since, for a given x 1 ∈ S, there clearly exist (iii)] at x lie in k X × f ∈ K X [at least after possibly passing to an appropriate connected ﬁnite étale covering of X] that have a zero or pole at x 1 but not at some other x ∈ S, this observation allows us to recover the canonical discrete structure [cf. [Mzk8], Deﬁ- nition 4.1, (iii); the proof of [Mzk8], Theorem 4.10] on the decomposition groups in cl Π c-ab U S [where S 1 ⊆ X is an arbitrary ﬁnite subset containing S, which corresponds, 1 42 SHINICHI MOCHIZUKI say, to a subset T 1 ⊆ Y cl that contains T ] at arbitrary points [i.e., arbitrary “x 1 ”] of S. Thus, by applying this canonical discrete structure [as in the proof of [Mzk8], Theorem 4.10], we may recover the subset × × f · k X ⊆ f · (k X ) ∧ × for arbitrary f ∈ K X [i.e., even f that have a zero or pole at every point of S] as the subset of elements for which the restriction to each point x of S either lies in × × ∧ k X ⊆ (k X ) or [when the element in question has a zero or pole at x] is compatible with the canonical discrete structure at x. Since this characterization of the subset × × ∧ ⊆ f · (k X ) is manifestly compatible [in light of the Green-compatibility f · k X ∼ c-ab assumption on α] with the isomorphisms Π c-ab U S 1 → Π V T 1 induced by α, we thus conclude that the isomorphism ∧ ∼ ∧ × × K X · (k X ) → K Y × · (k Y × ) ∧ × × × of Theorem 2.5, (ii), maps the subset K X ⊆ K X · (k X ) onto the subset K Y × ⊆ ∧ K Y × · (k Y × ) , as desired. Remark 2.7.1. Suppose, in the situation of Corollary 2.7, that S = X cl , T = Y cl . Then unlike the situation discussed in [Tama], one has the freedom to evaluate functions at arbitrary points of the entire sets X cl , Y cl , as opposed to just certain restricted subsets S ⊆ X cl , T ⊆ Y cl . Thus, instead of applying [Tama], Lemma 4.7, one may instead apply the somewhat easier argument implicit in [Uchi], §3, Lemmas 8-11 [which is used to treat the function ﬁeld case]. Thus, in light of Remark 2.6.1 [together with the portion of Theorem 1.16, (i), concerning the preservation of decomposition groups of cusps], Corollary 2.7 implies the following result, in the aﬃne case: Corollary 2.8. (Point-theoretic Isomorphisms in the Aﬃne Case) Let U , V be aﬃne hyperbolic curves over a ﬁnite or nonarchimedean local ﬁeld. Suppose that Σ = Primes. Write Δ U (respectively, Δ V ) for the maximal cuspidally pro-Σ † quotient of the maximal pro-Σ quotient of the tame geo- metric fundamental group of U (respectively, V ) [where “tame” is with respect to the complement of U (respectively, V ) in its canonical compactiﬁcation], and Π U (respectively, Π V ) for the corresponding quotient of the étale fundamental group of U (respectively, V ). Then any point-theoretic isomorphism ∼ β : Π U → Π V arises from a uniquely determined commutative diagram of schemes ∼ U ⏐ ⏐ → U → ∼ V ⏐ ⏐ V ABSOLUTE ANABELIAN CUSPIDALIZATIONS 43 in which the horizontal arrows are isomorphisms; the vertical arrows are the pro- ﬁnite étale coverings determined by the proﬁnite groups Π U , Π V . Remark 2.8.1. In light of the results of [Tama] [cf. Remarks 1.18.1, 1.18.2], Corollary 2.8 is only truly of interest in the case of nonarchimedean local ﬁelds. Deﬁnition 2.9. Suppose that k is a nonarchimedean local ﬁeld. (i) A [necessarily aﬃne] hyperbolic curve U over k will be said to be of strictly Belyi type if it is deﬁned over a number ﬁeld and isogenous [cf. §0] to a hyperbolic curve of genus zero. (ii) A [necessarily aﬃne] hyperbolic curve U over k will be said to be of Belyi type if it is deﬁned over a number ﬁeld, and, moreover, for some positive integer m, there exists a ﬁnite sequence U = U 1 U 2 . . . U m−1 U m of hyperbolic orbicurves [cf. §0] U j such that U m is a tripod [cf. §0], and, moreover, for each j = 1, . . . , m − 1, U j+1 is related to U j in one of the following ways: (a) there exists a ﬁnite étale morphism U j+1 → U j [i.e., “U j+1 is a ﬁnite étale covering of U j ”]; (b) there exists a ﬁnite étale morphism U j → U j+1 [i.e., “U j+1 is a ﬁnite étale quotient of U j ”]; (c) there exists an open immersion U j → U j+1 [i.e., in the terminology of [Mzk8], “U j+1 is a [hyperbolic] partial compactiﬁcation of U j ”]; (d) there exists a partial coarsiﬁcation morphism [cf. §0] U j → U j+1 [i.e., “U j+1 is a partial coarsiﬁcation of U j ”]. (iii) A [necessarily aﬃne] hyperbolic curve U over k will be said to be of quasi- Belyi type if it is deﬁned over a number ﬁeld and admits a connected ﬁnite étale covering V → U such that V admits a [not necessarily ﬁnite or étale!] dominant morphism V → W to a tripod W . Remark 2.9.1. It is immediate that every hyperbolic curve of strictly Belyi type is also of Belyi type [as the terminology suggests]. Moreover, one veriﬁes easily by “induction on m” [where “m” is as in Deﬁnition 2.9, (ii)] that every hyperbolic curve of Belyi type is also of quasi-Belyi type [as the terminology suggests]. It is not diﬃcult to see that there exist [multiply] punctured elliptic curves that are of Belyi type, but not of strictly Belyi type [cf. Remark 2.13.2 below]. On the other hand, it is not clear to the author at the time of writing whether or not there exist hyperbolic curves of quasi-Belyi type that are not of Belyi type. 44 SHINICHI MOCHIZUKI Remark 2.9.2. Hyperbolic curves of strictly Belyi type are precisely the sort of curves considered in [Mzk8], Corollaries 2.8, 3.2. Remark 2.9.3. The author would like to thank A. Tamagawa for useful discus- sions concerning Deﬁnition 2.9, (ii), especially Deﬁnition 2.9, (ii), (d). Proposition 2.10. (Decomposition Groups of Curves of Quasi-Belyi Type) Let U (respectively, V ) be a hyperbolic curve over a nonarchimedean local ﬁeld. Denote the base ﬁeld of U (respectively, V ) by k U (respectively, k V ), the étale fundamental group of U (respectively, V ) by Π U (respectively, Π V ) [i.e., “we take Σ = Primes”]. Let ∼ β : Π U → Π V be an isomorphism of proﬁnite groups. Then: (i) If U is of quasi-Belyi type, then the closed points of “DLoc-type” [in the sense of [Mzk8], Deﬁnition 2.4] are p U -adically dense [where p U is the residue characteristic of k U ] in U (k U ). (ii) If U is of quasi-Belyi type, then β maps every decomposition group of a closed point of U isomorphically onto a decomposition group of a closed point of V . (iii) If both U , V are of quasi-Belyi type, then β is point-theoretic. (iv) If U is of Belyi type, then so is V . Proof. The proof of assertion (i) is similar to the proof of [Mzk8], Corollary 2.8: That is to say, in the terminology of loc. cit., it follows formally from the fact that U is of quasi-Belyi type that the “algebraic” closed points [i.e., closed points deﬁned over a number ﬁeld, which are manifestly p U -adically dense in U (k U )] of U are of “DLoc-type” [cf. the proof of [Mzk8], Corollary 2.8]: Indeed, it suﬃces to consider the following commutative diagram of hyperbolic curves, whose existence follows from the assumption that U is of quasi-Belyi type: U ←− V ⏐ ⏐ −→ W ⏐ ⏐ V −→ → U −→ U W Here, the “hooked arrow →” is an open immersion; all of the “non-hooked arrows” except for V → W , V → W are ﬁnite étale morphisms; V → W , V → W are dominant; the ﬁnite étale morphism U → U is obtained by a base-change to a ﬁnite extension of the base ﬁeld k U ; and W is a tripod [so W → W is a “Belyi map”]. Note that the composite arrow V → W → U → U may be thought of as an arrow in the category DLoc k U (U ) of [Mzk8], §2. Observe, moreover, that the arrow W → U may be chosen to have arbitrarily designated algebraic closed points in the complement of its image. Thus, we conclude that this diagram exhibits the ABSOLUTE ANABELIAN CUSPIDALIZATIONS 45 [arbitrarily designated] algebraic closed points in the complement of the image of W → U → U as points of DLoc-type, as desired. This completes the proof of assertion (i). In light of assertion (i) [applied to the various connected ﬁnite étale coverings of U ], the proof of assertion (ii) is entirely similar to the proof of [Mzk8], Corollary 3.2: That is to say, by [Mzk8], Corollary 2.5, it follows that β maps decomposition groups of DLoc-type of U to decomposition groups of DLoc-type of V . Thus, assertion (ii) follows by applying [Mzk8], Lemma 3.1 [where the density statement of assertion (i) concerning points of DLoc-type allows one to replace the “algebraicity” condition of [Mzk8], Lemma 3.1, (iii), by the condition that the points in question be of DLoc- type]. Finally, assertion (iii) follows formally from assertion (ii) [and Proposition 2.3, (ii)]. Finally, we consider assertion (iv). First, I claim that by applying the iso- morphism β [and thinking of hyperbolic orbicurves as being represented by their associated étale fundamental groups], one may transform the sequence U = U 1 U 2 . . . U m−1 U m of Deﬁnition 2.9, (ii), into a sequence V = V 1 V 2 . . . V m−1 V m that also satisﬁes the conditions of Deﬁnition 2.9, (ii), in such a way that we also ∼ obtain compatible isomorphisms β j : Π U j → Π V j [where j = 1, . . . , m; β 1 = β]. Indeed, we reason by induction on m. If [for j = 1, . . . , m − 1] U j+1 is related to U j as in (a) [of Deﬁnition 2.9, (ii)], then it is immediate [by thinking in terms of open subgroups of Π U j , Π V j ] that one may construct [from V j ] a V j+1 related to V j as in (a). If U j+1 is related to U j as in (b) (respectively, (c)), then it follows from [Mzk6], Theorem 2.4 (respectively, [Mzk8], Theorem 1.3, (iii) [cf. also [Mzk8], Theorem 2.3]), that one may construct [from V j ] a V j+1 related to V j as in (b) (respectively, (c)). If U j+1 is related to U j as in (d), then Π U j+1 is obtained from Π U j by forming the quotient of Π U j by the closed normal subgroup of Π U j generated by some ﬁnite collection of elements of Δ U j that belong to the decomposition groups of points of U j in Δ U j . Thus, by Lemma 2.11, (v), below, we conclude that the quotient Π U j Π U j+1 determines a quotient Π V j Π V j+1 that corresponds to a partial coarsiﬁcation V j → V j+1 , as desired. Finally, if U m is a tripod, the existence ∼ of the isomorphism Π U m → Π V m implies that V m is also a tripod [cf. [Mzk5], Lemma 1.3.9]. This completes the proof of the claim. Thus, to complete the proof of assertion (iv), it suﬃces to verify that V is deﬁned over a number ﬁeld. But observe that since U is deﬁned over a number ﬁeld, there exists a diagram of hyperbolic curves [i.e., in essence, a “Belyi map”] U m ←− U m → U −→ U where the “hooked arrow →” is an open immersion; the “non-hooked arrows” are ﬁnite étale morphisms; and the ﬁnite étale morphism U → U is obtained by 46 SHINICHI MOCHIZUKI a base-change to a ﬁnite extension of the base ﬁeld k U . Now the isomorphisms ∼ ∼ Π U m → Π V m , Π U → Π V allow us to transform [cf. [Mzk8], Theorem 2.3 and its proof] this diagram into a similar diagram V m ←− V m → V −→ V whose existence [since V m is also a tripod!] shows that V is also deﬁned over a number ﬁeld, as desired. This completes the proof of assertion (iv). Remark 2.10.1. Note that the essential reason that the author is unable to prove the stronger statement of Proposition 2.10, (iv), in the quasi-Belyi case is that, in the notation of the proof of Proposition 2.10, (i), it is unclear how to construct [at the level of arithmetic fundamental groups] the dominant morphism V → W from V . That is to say, unlike the situation involving the operations of Deﬁnition 2.9, (ii), (a), (b), (c), (d), it is by no means clear how to construct, via purely group- theoretic operations, the quotient of an arithmetic fundamental group arising from an arbitrary dominant morphism. Lemma 2.11. (Finite Subgroups of Fundamental Groups of Hyperbolic Orbicurves) Let W be a hyperbolic orbicurve over an algebraically closed ﬁeld of characteristic zero; Σ W a nonempty set of prime numbers. Denote the maximal pro-Σ W quotient of the étale fundamental group of W by Δ W ; suppose that W admits a ﬁnite étale covering by a hyperbolic curve that arises from an open subgroup of Δ W . Let A ⊆ Δ W (respectively, B ⊆ Δ W ) be the decomposition group [well-deﬁned up to conjugation in Δ W ] of a closed point w A (respectively, w B ) of W ; suppose that w A = w B . Then: (i) A, B are cyclic. (ii) A Δ W . B = {1}. In particular, if A = {1}, then A is normally terminal in (iii) The order of every ﬁnite cyclic closed subgroup C ⊆ Δ W divides the order of W [cf. §0]. (iv) Every ﬁnite nontrivial closed subgroup C ⊆ Δ W is contained in a decomposition group of a unique closed point of W . (v) The nontrivial decomposition groups of closed points of W may be charac- terized as the maximal ﬁnite nontrivial closed subgroups of Δ W . Proof. Assertion (i) follows immediately from the well-known [and easily veriﬁed] fact that the absolute Galois group of a complete discrete valuation ﬁeld with algebraically closed residue ﬁeld of characteristic zero is cyclic. Next, we consider assertion (ii). Let C ⊆ A B be a subgroup of prime order l ∈ Σ W . Now consider a normal open subgroup H ⊆ Δ W such that the covering W H → W determined by H is a hyperbolic curve. Note that this implies that ABSOLUTE ANABELIAN CUSPIDALIZATIONS 47 A H = B H = C H = {1} [cf., e.g., assertion (iii), which will be proven below without applying the present assertion (ii)]. Write W H → W C → W for the covering determined by the open subgroup C · H ⊆ Δ W . Observe that there exist , w B of W C that lift w A , w B , respectively, and whose decomposition closed points w A groups [well-deﬁned up to conjugation in C · H] are equal to C. Note that since W H is a hyperbolic curve, and C is of prime order l, it follows that the order of every closed point of W C is equal to either 1 or l. Now if W C is aﬃne, then let v be a cusp of W C . If W C is proper and admits ≥ 3 points of order l, then let v be a point of W C of order l such that v = w A , w B . Note that if W C is proper and admits ≤ 2 points of order l, then it follows from the hyperbolicity assumption that the coarsiﬁcation of W C is a proper smooth curve of genus ≥ 1; thus, by replacing H by an appropriate open subgroup of H, one veriﬁes immediately that one may assume without loss of generality that either W C is aﬃne or W C admits ≥ 3 points of order l. Now observe that W C admits a ﬁnite étale cyclic covering W C → W C of degree l which is étale over the compactiﬁcation of the coarsiﬁcation of W C , except over the , points in the compactiﬁcation of the coarsiﬁcation of W C corresponding to v, w B over which W C is totally ramiﬁed. In particular, it follows that any point of W C lying over w A (respectively, w B ) is of order l (respectively, 1), thus contradicting the observation that the decomposition groups [well-deﬁned up to conjugation in C · H] of w A , w B are equal to C. This completes the proof that A B = {1}. By applying this fact to arbitrary ﬁnite étale coverings of W , it follows formally [cf. Proposition 2.6, (ii)] that A is normally terminal in Δ W , whenever A = {1}. Next, we consider assertion (iii). Denote the order of W by n. Now if C ⊆ Δ W is a nontrivial ﬁnite cyclic closed subgroup, then there exists a normal open subgroup def N ⊆ Δ W such that C N = {1}. In particular, it follows that if we take H = C ·N [so H ⊆ Δ W is an open subgroup], then the natural map C → H ab is injective. On the other hand, if we denote by W H → W the covering determined by H, then it is clear that the order of W H divides n, hence that H ab is an extension of a torsion- free proﬁnite abelian group by a ﬁnite abelian group annihilated by n. Thus, we conclude from the injection C → H ab that the order of C divides n, as desired. This completes the proof of assertion (iii). Next, we consider assertion (iv). First, let us observe that uniqueness follows formally from assertion (ii). Next, let us verify assertion (iv) under the further assumption that C is solvable. By induction on the order of C, we may assume that [at least] one of the following conditions is satisﬁed: (a) C is an extension of a group of prime order by a nontrivial subgroup C 1 ⊆ C which is contained in the decomposition group A; (b) C is of prime order l ∈ Σ W . If (a) is satisﬁed, then by replacing W by a ﬁnite étale covering of W determined by a suitable open subgroup containing C, we may assume that (C 1 ⊆) A ⊆ C. Thus, if A = C, then A = C 1 is normal in C. But this implies, by the normal terminality portion of assertion (ii), that A = C, a contradiction. Thus, (a) implies that C ⊆ A. If (b) is satisﬁed, then we argue as follows: Observe that by assertion (iii), every open subgroup H ⊆ Δ W that contains C determines a ﬁnite étale covering W H → W such that the order of W H is divisible by l. Write Stack l (W H ) for the set of closed points of W H whose order is divisible by l. Now observe that 48 SHINICHI MOCHIZUKI since the order of W H is divisible by the prime number l, it follows that Stack l (W H ) is nonempty. Since the set Stack l (W H ) is ﬁnite and nonempty, we thus conclude that, if we allow H to vary [among open subgroups H ⊆ Δ W that contain C], then the inverse limit lim ←− Stack l (W H ) H is nonempty. But, unraveling the deﬁnitions, this means precisely that C contains the decomposition group D associated to some compatible system of points of the sets Stack l (W H ). Since D is of order divisible by l, we thus conclude that D = C, as desired. This completes the proof of assertion (iv) for C solvable. On the other hand, a well-known theorem from the theory of ﬁnite groups asserts that a ﬁnite group in which every Sylow subgroup is cyclic is solvable [cf. [Scott], p. 356]. Thus, in light of assertion (i), we conclude that assertion (iv) for C solvable implies assertion (iv) for C arbitrary. Finally, we observe that assertion (v) follows formally from assertions (ii), (iv). Remark 2.11.1. The author would like to thank A. Tamagawa for informing him of Lemma 2.11 and, in particular, of the theorem on ﬁnite groups that was applied in the proof of Lemma 2.11, (iv). We are now ready to state the following “absolute p-adic version of the Grothen- dieck Conjecture” for hyperbolic curves of Belyi or quasi-Belyi type: Corollary 2.12. (Curves of Belyi or Quasi-Belyi Type) Let U (respectively, V ) be a hyperbolic curve over a nonarchimedean local ﬁeld. Denote the base ﬁeld of U (respectively, V ) by k U (respectively, k V ), the étale fundamental group of U (respectively, V ) by Π U (respectively, Π V ) [i.e., “we take Σ = Primes”]. Suppose further that at least one of the following conditions holds: (a) both U and V are of quasi-Belyi type; (b) either U or V [but not necessarily both!] is of Belyi type. Then any isomorphism of proﬁnite groups ∼ β : Π U → Π V arises from a uniquely determined commutative diagram of schemes ∼ U ⏐ ⏐ → U → ∼ V ⏐ ⏐ V ABSOLUTE ANABELIAN CUSPIDALIZATIONS 49 in which the horizontal arrows are isomorphisms; the vertical arrows are the pro- ﬁnite étale coverings determined by the proﬁnite groups Π U , Π V . Proof. In light of Proposition 2.10, (iii), (iv) [cf. also Remark 2.9.1], Corollary 2.12 follows formally from Corollary 2.8. Remark 2.12.1. Note that in the proof of Proposition 2.10, Corollary 2.12, it is necessary, in the quasi-Belyi case, to apply the full “Hom version” of [Mzk4], Theorem A. This diﬀers from the situation of [Mzk8], Corollaries 2.8, 3.2 — i.e., where one only treats hyperbolic curves of strictly Belyi type — or, indeed, of the portion of Proposition 2.10, Corollary 2.12, that concerns curves of Belyi type, in which the “isomorphism version” of [Mzk4], Theorem A, suﬃces [cf. [Mzk8], Remark 2.8.1]. Thus, in the terminology of [Mzk6], Deﬁnition 3.7, the portion of Corollary 2.12 concerning hyperbolic curves of Belyi type admits the following formal consequence: Corollary 2.13. (Absoluteness of Curves of Belyi Type) Every hyperbolic curve of Belyi type over a nonarchimedean local ﬁeld is absolute. Remark 2.13.1. It is interesting to note that the essential property that underlies the absoluteness of Corollary 2.13 is the existence of a Belyi map [since the curve is deﬁned over a number ﬁeld], which, in the context of the theory of [Mzk8], §2, may be regarded as a sort of endomorphism of the curve. From this point of view, Corollary 2.13 is reminiscent of [Mzk6], Corollary 3.8, which states that the “canonical curves” of p-adic Teichmüller theory are absolute. Indeed, from the point of view of the theory of [Mzk2], this canonicality may be regarded as the existence of a sort of “Frobenius endomorphism” of the curve. It is also interesting to note that both of these results assert that every member of some countable collection of nonarchimedean hyperbolic curves is absolute. Remark 2.13.2. In the context of Remark 2.13.1, it is interesting to note that, unlike the canonical curves discussed in [Mzk6], §3, the set of points determined by the hyperbolic curves of strictly Belyi type fails, for all pairs (g, r) such that 2g − 2 + r ≥ 3, g ≥ 1, to be Zariski dense in the moduli stack of hyperbolic curves of type (g, r). Indeed, this follows immediately from [Mzk1], Theorem B. On the other hand, it is not clear to the author at the time of writing whether or not the set of points determined by the hyperbolic curves of Belyi (respectively, quasi-Belyi) type is Zariski dense in the moduli stack of hyperbolic curves of type (g, r) [when, say, 2g − 2 + r ≥ 3, g ≥ 2]. Note, however, that when g = 0, 1, [one veriﬁes easily that] every hyperbolic curve of type (g, r) that is deﬁned over a number ﬁeld is automatically of Belyi type. 50 SHINICHI MOCHIZUKI Section 3: Maximal Pro-l Cuspidalizations In this §, we apply the theory of the weight ﬁltration [cf. [Kane], [Mtm]], together with various generalities concerning free Lie algebras [cf. the Appendix], to construct, in the ﬁnite ﬁeld case, “maximal cuspidally pro-l cuspidalizations” [cf. Theorem 3.10], whose existence implies, under quite general conditions [cf. Corollary 3.11 below], that an isomorphism “α” as in Theorem 1.16, (iii), is always totally globally Green-compatible. In the following discussion, we maintain the notation of §2, and assume further throughout the present §3 that we are in the ﬁnite ﬁeld case. Deﬁnition 3.1. Let l be a prime number; G, H, A topologically ﬁnitely generated pro-l groups; φ : H → A a [continuous] homomorphism. Suppose further that A is abelian, and that G is an l-adic Lie group [cf., e.g., [Serre], Chapter V, §7, §9, for basic facts concerning l-adic Lie groups]. (i) We shall refer to as the φ-central ﬁltration on H the ﬁltration deﬁned as follows: def H(1) = H def H(2) = Ker(φ) def H(m) = the subgroup topologically generated by the commutators [H(a), H(b)], where a + b = m, ∀ m ≥ 3 Thus, in words, this ﬁltration on H is the “fastest decreasing central ﬁltration among those central ﬁltrations whose top quotient factors through φ”. We shall say that H is φ-nilpotent if H(m) = {1} for suﬃciently large m. If H is φ-nilpotent when φ is taken to be the natural surjection H H ab to its abelianization H ab , then we shall say that H is nilpotent. In the following, for a, b, n ∈ Z such that 1 ≤ a ≤ b, n ≥ 1, we shall write def H(a/b) = H(a)/H(b) and def Gr(H)(n) = def H(m/m + 1) ⊆ Gr(H) = Gr(H)(1) m≥n def Gr(H)(a/b) = Gr(H)(a)/Gr(H)(b) and append a subscript Q l (respectively, F l ) to these objects to denote the result of tensoring over Z l with Q l (respectively, F l ). Thus, Gr(H), Gr Q l (H), Gr F l (H) are graded Lie algebras over Z l , Q l , F l , respectively; Gr(H)(n) ⊆ Gr(H) is a [Lie algebra-theoretic] ideal. Also, if Z a ≥ 1, then we shall write: def H(a/∞) = ← lim − H(a/b) b [where b ranges over the integers ≥ a + 1]. ABSOLUTE ANABELIAN CUSPIDALIZATIONS 51 (ii) We shall denote by Lie(G) the Lie algebra over Q l determined by G. If G is nilpotent, then Lie(G) is a nilpotent Lie algebra over Q l , hence determines a connected, unipotent linear algebraic group Lin(G), which we shall refer to as the linear algebraic group associated to G. In this situation, there exists [cf., e.g., Remark 3.3.2 below] a natural [continuous] homomorphism [with open image] G → Lin(G)(Q l ) [from G to the l-adic Lie group determined by the Q l -valued points of Lin(G)] which is uniquely determined [since Lin(G) is connected and unipotent!] by the condition that it induce the identity morphism on the associated Lie algebras. In the situation of (i), if Z a ≥ 1, then we shall write: def lim Lie(H(a/∞)) = ← − Lie(H(a/b)); b def Lin(H(a/∞)) = ← lim − Lin(H(a/b)) b [where b ranges over the integers ≥ a + 1; we recall that it is well-known [or easily veriﬁed] that each H(a/b) is an l-adic Lie group]. Now let us ﬁx a prime number l ∈ Σ † . For S ⊆ X(k) a ﬁnite subset, let us denote by (l) (l) Δ U S Δ U S ; Δ X Δ X the maximal pro-l quotients and by (l) Π U S Π U S ; (l) Π X Π X (l) (l) the quotients of Π U S , Π X by the kernels of Δ U S Δ U S , Δ X Δ X . [Here, we recall that Δ U S , Π U S are as deﬁned in Proposition 1.8, (ii), (iii).] Also, for x ∈ X cl , let us write (l) (l) D x (l) [U S ] ⊆ Π U S ; I x (l) [U S ] ⊆ Δ U S for the images of D x [U S ], I x [U S ] [notation as in Proposition 2.1], respectively, in (l) Π U S . Note that we have a natural surjection: (l) (l) (l) Δ U S Δ X (Δ X ) ab (l) The cup product on the group cohomology of Δ X determines an isomorphism [cf. Proposition 1.3, (ii)] (l) (l) ∼ (l) Hom((Δ X ) ab , M X ) → (Δ X ) ab (l) def [where we write M X = M X ⊗ Z l ], hence a natural G k -equivariant injection (l) (l) M X → ∧ 2 (Δ X ) ab (l) whose image we denote by I cup . 52 SHINICHI MOCHIZUKI Deﬁnition 3.2. We shall refer to the central ﬁltration (l) {Δ U S (m)} (l) (l) (l) on Δ U S with respect to the natural surjection Δ U S (Δ X ) ab as the weight ﬁltra- (l) tion on Δ U S [cf., e.g., [Mtm], §3, p. 200]. def Proposition 3.3. (Freeness and Centralizers) Let x ∈ S. Write S x = S\{x}; r for the cardinality of S, g for the genus of X. For x ∈ S, let ζ x be a (l) generator of I x [U S ]. By abuse of notation, we shall also denote by ζ x the image (l) of ζ x in Δ U S (2/3). Then: (l) (i) Gr(Δ U S ) is a free Lie algebra over Z l [hence, in particular, is torsion-free as a Z l -module] which is freely generated by 2g elements (l) α 1 , . . . , α g , β 1 , . . . , β g ∈ Δ U S (1/2) together with the ζ x ∈ Δ U S (2/3), for x ∈ S x . Alternatively, for an appropriate (l) (l) choice of the elements ζ x , Gr(Δ U S ) is the quotient of the free Lie algebra generated (l) by α 1 , . . . , α g , β 1 , . . . , β g , together with the ζ x ∈ Δ U S (2/3), for x ∈ S, by the single relation: g ζ x + [α n , β n ] = 0 x ∈S n=1 At a more intrinsic level, this relation is a generator of the image of the natural G k -equivariant morphism (l) M X → (l) I x [U S ] (l) ⊕ I cup x ∈S (l) ∼ (l) (l) ∼ (l) [determined by the various natural isomorphisms M X → I x [U S ], M X → I cup ]], (l) whose codomain maps to Gr(Δ U S ) via the natural G k -equivariant morphism (l) I x [U S ] x ∈S (l) (l) ⊕ I cup → Δ U S (2/3) (l) (l) [determined by the natural inclusions I x [U S ] → Δ U S (2/3) and the bracket opera- (l) (l) tion ∧ 2 (Δ X ) ab → Δ U S (2/3)]. (ii) Let ξ be any of the elements α 1 , . . . , α g , β 1 , . . . , β g ; ζ x , where x ∈ S x , (l) (l) of (i). Then the centralizer in Gr Q l (Δ U S ) of [the image of ] ξ [in Gr Q l (Δ U S )] is (l) equal to Q l · ξ. In particular, the Lie algebra Gr Q l (Δ U S ) is center-free. ABSOLUTE ANABELIAN CUSPIDALIZATIONS 53 (l) (iii) Let ξ be as in (ii). Then for m ≥ 1, the centralizer in Δ U S (1/m + 2) of (l) (l) [the image of ] ξ [in Δ U S (1/m + 2)] is contained in the subgroup of Δ U S (1/m + 2) (l) generated by [the image of ] ξ and Δ U S (m/m + 2). (iv) Let S ∗ ⊆ S be a subset of S. Write (l) (l) New S ∗ ⊆ Gr(Δ U S ) for the sub-Lie algebra over Z l generated by the image of the restriction (l) (l) (l) I x [U S ] ⊆ I x [U S ] → Δ U S (2/3) x ∈S ∗ x ∈S to the direct summands indexed by elements of S ∗ of the morphism of (i), and (l) def (l) New S ∗ (a) = Gr(Δ U S )(a) (l) (l) def (l) (l) New S ∗ ; New S ∗ (a/b) = New S ∗ (a)/New S ∗ (b) for a, b ∈ (l) Z such that 1 ≤ a ≤ b. Then, in the notation of (i), New S ∗ is a free Lie algebra over Z l generated by the elements ζ x , for x ∈ S ∗ . Moreover, the [“new” and “co-new”] Z l -modules (l) New S ∗ (a/b); def (l) (l) (l) Cnw S ∗ (a/b) = Gr(Δ U S )(a/b)/New S ∗ (a/b) tor,(l) are free. In the following discussion, we shall write New S ∗ Q/Z. def (l) (a/b) = New S ∗ (a/b)⊗ Proof. Assertion (i) (respectively, (ii)) is, in essence, the content of [Kane], Propo- sition 1 (respectively, Proposition A.1, (ii), (iii)). Assertion (iii) follows formally from assertion (ii). Finally, we consider assertion (iv). By Proposition A.1, (iii), it follows that any free Lie algebra over F l with ≥ 2 generators is center-free. Thus, let M be the module determined by any faithful representation [e.g., when the car- dinality of S ∗ is ≥ 2, the adjoint representation] of the free Lie algebra F over F l in the formal generators ζ x , where x ∈ S ∗ . Now observe that we obtain an def (l) action of Gr F l (Δ U S ) on M = M ⊕ M as follows: We let α 2 , . . . α g ; β 2 , . . . β g ; ζ x , def where x ∈ S 0 = S\S ∗ , act by multiplication by 0 on M . We let α 1 , β 1 act on M = M ⊕ M via the matrices ζ x 0 0 0 x ∈S ∗ ; −1 0 0 0 respectively. Finally, we let ζ x , where x ∈ S ∗ , act on M via the following matrix: 0 ζ x 0 −ζ x Thus, [by assertion (i)] M determines a representation of Gr F l (Δ U S ) whose re- (l) (l) (l) striction to the image of New S ∗ ⊗ Z l F l in Gr F l (Δ U S ) determines [via the natural 54 SHINICHI MOCHIZUKI (l) surjection F New S ∗ ⊗ Z l F l ] a faithful representation of F . Thus, we conclude that (l) (l) the natural surjection F New S ∗ ⊗ Z l F l is an isomorphism, and that New S ∗ ⊗ Z l F l (l) injects into Gr F l (Δ U S ). Assertion (iv) now follows formally. Remark 3.3.1. The author wishes to thank A. Tamagawa for pointing out to him the content of Proposition 3.3, (i). Remark 3.3.2. One way to verify the existence of the homomorphism “G → Lin(G)(Q l )” of Deﬁnition 3.1, (ii), is to think of G as a quotient of a free pro-l group of ﬁnite even rank F , whose associated “Gr Q l (−)” is a center-free free Lie algebra [cf. Proposition 3.3, (i), (ii), in the case of r = 1], hence determines an [inﬁnite-dimensional, over Q l ] faithful [cf. Proposition 3.3, (iii)] unipotent represen- tation [i.e., the adjoint representation — cf. the proof of Proposition 3.3, (iv)] of F . More precisely, by Proposition 3.3, (iii), it follows that there exists a unipotent linear representation ρ F : F → GL(V ) on a ﬁnite-dimensional Q l -vector space V such that Ker(ρ F ) ⊆ Ker(F G). But this implies that F G factors through a quotient F Q G such that Q is nilpotent and admits an injective homo- morphism of topological groups ρ Q : Q → Q alg (Q l ) [induced by ρ F ], where Q alg is a connected, unipotent algebraic group over Q l , such that ρ Q is a local isomor- phism, and Ker(ρ Q ) ⊆ Ker(Q G). Thus, ρ Q determines a structure of l-adic Lie group on Q such that the morphism Lie(ρ Q ) induced by ρ Q on Lie algebras is an isomorphism. Moreover, the morphism induced by Q G on Lie algebras factors through Lie(ρ Q ), thus determining a homomorphism of [connected, unipotent] al- gebraic groups Q alg → Lin(G) such that the resulting composite homomorphism Q → Q alg (Q l ) → Lin(G)(Q l ) factors [cf. the induced morphisms on Lie algebras, together with the fact that Lin(G)(Q l ) has no torsion!] though G, thus yielding a homomorphism G → Lin(G)(Q l ), as desired. Next, let us ﬁx an x ∗ ∈ S, as well as a choice of decomposition group D x ∗ [U S ] ⊆ Π U S [i.e., among the various Π U S -conjugates of this subgroup] associated to x ∗ . [Thus, D x ∗ [U S ] determines a speciﬁc subgroup [i.e., not just a conjugacy class of subgroups] (l) (l) D x ∗ [U S ] ⊆ Π U S .] Recall that the natural exact sequences 1 → I x ∗ [U S ] → D x ∗ [U S ] → G k → 1; 1 → I x (l) [U S ] → D x (l) ∗ [U S ] → G k → 1 ∗ split. [Indeed, extracting roots of any local uniformizer of X at x ∗ determines such a splitting — cf., e.g., the discussion at the beginning of [Mzk8], §4.] In the following discussion, we shall ﬁx a splitting G k → D x ∗ [U S ] ABSOLUTE ANABELIAN CUSPIDALIZATIONS 55 of this exact sequence. Thus, this splitting determines a natural action of G k [by (l) conjugation] on Δ U S , hence also on def (l) (l) def (l) Lin U S (a/b) = Lin(Δ U S (a/b))(Q l ); (l) Lie U S (a/b) = Lie(Δ U S (a/b)) (l) Gr Q l (Δ U S )(a/b) [where a, b ∈ Z; 1 ≤ a ≤ b]. Write F k ∈ G k for the Frobenius element of G k . In the following, we shall denote the cardinality of k by q k . (Galois Invariant Splitting) Let a, b ∈ Z, 1 ≤ a ≤ b. Proposition 3.4. (l) (i) The eigenvalues of the action of F k on Lie U S (a/a + 1) are algebraic num- a/2 bers all of whose complex absolute values are equal to q k [i.e., “of weight a”]. (ii) There is a unique G k -equivariant isomorphism of Lie algebras ∼ (l) (l) Lie U S (a/b) → Gr Q l (Δ U S )(a/b) ∼ (l) (l) which induces the identity isomorphism Lie U S (c/c + 1) → Gr Q l (Δ U S )(c/c + 1), for all c ∈ Z such that a ≤ c ≤ b − 1. (l) (iii) The isomorphism of (ii) together with the natural inclusions I x [U S ] → (l) (l) Δ U S for x ∈ S [which are well-deﬁned up to Δ U S -conjugation] determine a G k - equivariant morphism I x (l) [U S ] ⊗ Q l (l) (l) ⊕ Lie U S (1/2) → Lie U S (1/∞) x∈S (l) which exhibits, in a G k -equivariant fashion, Lie U S (1/∞) as the quotient of the completion [with respect to the ﬁltration topology] of the free Lie algebra generated by the ﬁnite dimensional Q l -vector space I x (l) [U S ] ⊗ Q l (l) ⊕ Lie U S (1/2) x∈S (l) [equipped with a natural grading, hence also a ﬁltration, by taking the I x [U S ] ⊗ (l) Q l to be of weight 2, Lie U S (1/2) to be of weight 1], by the single relation deter- mined by the image of the morphism (l) M X ⊗ Q l → x∈S (l) I x (l) [U S ] ⊗ Q l ⊕ (I cup ⊗ Q l ) 56 SHINICHI MOCHIZUKI of Proposition 3.3, (i), tensored with Q l . (l) that (l) (iv) For each g ∈ Lin U S (1/∞), there exists a unique h ∈ Lin U S (1/∞) such F k ◦ Inn g = Inn h ◦ F k ◦ Inn h −1 (l) [where “Inn” denotes the inner automorphism of Lin U S (1/∞) deﬁned by conjuga- (l) tion by the subscripted element]. Moreover, when g lies in the image of I x ∗ ⊗ Q l (l) [which is stabilized by the action of F k ], h also lies in the image of I x ∗ ⊗ Q l . Proof. Assertion (i) follows immediately from the “Riemann hypothesis for abelian varieties over ﬁnite ﬁelds” — cf., e.g., [Mumf], p. 206. Assertion (ii) (respec- tively, (iii); (iv)) follows formally from assertion (i) (respectively, and Proposition 3.3, (i); and successive approximation of h with respect to the natural ﬁltration (l) (l) Lin U S (a/∞) ⊆ Lin U S (1/∞)). Next, let S ∗ ⊆ S def be a subset such that x ∗ ∈ S ∗ ; S 0 = S\S ∗ . In the following, we shall regard (l) Lin U S (a/b) as being equipped with its natural l-adic topology. Thus, G k acts con- (l) (l) tinuously on Lin U S (a/b), Lie U S (a/b), and we have natural G k -equivariant surjec- tions: (l) (l) (l) (l) Lin U S (a/b) Lin U S (a/b); Lie U S (a/b) Lie U S (a/b) 0 Let us write 0 (l) Lin U S /U S (a/b); 0 (l) Lie U S /U S (a/b) 0 for the kernels of these surjections. In the following, to simplify the notation, we shall often omit the superscript (l) from the objects “Lin (l) ”, “Lie (l) ”, “New (l) ”, “New tor,(l) ” introduced above and write: Lin U S (a/b); Lin U S /U S 0 (a/b); Lie U S (a/b); Lin U S 0 (a/b); Lie U S /U S 0 (a/b); Lie U S 0 (a/b) New S ∗ (a/b); New tor S ∗ (a/b) Also, we shall write: New Q S ∗ (a/b) = New S ∗ (a/b) ⊗ Q; def Note that, for Z def Δ Lie U S = Lin U S (1/∞) × Lin US (1/∞) Δ U S 0 0 b ≥ 1, we have a natural G k -equivariant inclusion ∼ Lin U S /U S 0 (b + 1/∞) → Lin U S /U S 0 (b + 1/∞) × {1} {1} → Lin U S (1/∞) × Lin US (1/∞) Δ U S 0 0 = Δ Lie U S whose image forms a normal subgroup of Δ Lie U S ; write Lie≤b Δ Lie U S Δ U S ABSOLUTE ANABELIAN CUSPIDALIZATIONS 57 for the quotient of Δ Lie U S by this normal subgroup. Also, we have a natural G k - equivariant [composite] inclusion ∼ Lie≤b+1 New Q S ∗ (b + 1/b + 2) → Lie U S /U S 0 (b + 1/b + 2) → Lin U S /U S 0 (b + 1/b + 2) → Δ U S whose image forms a normal subgroup of Δ Lie≤b+1 ; write U S Lie≤b+1 Lie≤b+ Δ U Δ U S S Lie≤b+1 for the quotient of Δ U by this normal subgroup. Thus, we have natural G k - S equivariant homomorphisms of topological groups: Lie≤b+ Δ U S → Δ Lie Δ Lie≤b Δ U S 0 U S Δ U S U S [the last three of which are easily veriﬁed to be surjective]. Moreover, forming the semi-direct product with G k [via the natural actions of G k ] yields topological groups and homomorphisms as follows: Lie≤b+ Π U S → Π Lie Π Lie≤b Π U S 0 U S Π U S U S Also, we note that we have natural exact sequences: 1 → Lin U S /U S 0 (1/∞) → Δ Lie U S → Δ U S 0 → 1 1 → Lin U S /U S 0 (1/∞) → Π Lie U S → Π U S 0 → 1 Deﬁnition 3.5. Lie≤b Lie≤b+ Lie ; Π Lie≤b ; Δ U ; Π Lie≤b+ ) (i) We shall refer to Δ Lie U S (respectively, Π U S ; Δ U S U S U S S as the [l-adic] Lie-iﬁcation (respectively, Lie-iﬁcation; Lie-iﬁcation, truncated to order b; Lie-iﬁcation, truncated to order b; Lie-iﬁcation, truncated to order b+; Lie- iﬁcation, truncated to order b+) of Δ U S (respectively, Π U S ; Δ U S ; Π U S ; Δ U S ; Π U S ) [over Δ U S 0 (respectively, Π U S 0 ; Δ U S 0 ; Π U S 0 ; Δ U S 0 ; Π U S 0 )]. (ii) Observe that it follows immediately from the deﬁnitions that, for Z we have natural exact sequences b ≥ 1, Lie≤b+1 Lie≤b+ → Δ U → 1 1 → New Q S ∗ (b + 1/b + 2) → Δ U S S Lie≤b+1 → Π Lie≤b+ → 1 1 → New Q S ∗ (b + 1/b + 2) → Π U S U S on which Π Lie≤b+1 acts naturally by conjugation. [Here, we note in passing that it U S is immediate from the deﬁnitions that the submodule New S ∗ (b + 1/b + 2) ⊆ New Q S ∗ (b + 1/b + 2) is contained in the image of Δ U S .] In particular, we obtain a natural inclusion: Lie≤b+1 New S ∗ (b + 1/b + 2) → Δ U (⊆ Π Lie≤b+1 ) U S S 58 SHINICHI MOCHIZUKI We shall refer to the quotients of Δ Lie≤b+1 , Π Lie≤b+1 by the image of this natural U S U S tor≤b+1 tor≤b+1 inclusion as the toral Lie-iﬁcations Δ U S , Π U S of Δ U S , Π U S [over Δ U S 0 , Π U S 0 ]. Thus, we have natural exact sequences tor≤b+1 Lie≤b+ → Δ U → 1 1 → New tor S ∗ (b + 1/b + 2) → Δ U S S tor≤b+1 → Π Lie≤b+ → 1 1 → New tor S ∗ (b + 1/b + 2) → Π U S U S acts naturally by conjugation. on which Π Lie≤b+1 U S (iii) Suppose that U S → U S 0 is a connected ﬁnite étale covering that arises 0 from an open subgroup Π U ⊆ Π U S 0 ; write X → X for the normalization of X in S 0 U S . Then we shall say that the [ramiﬁed] covering X → X is (S, S 0 , Σ)-admissible 0 if every closed point of X that lies over a point of S is rational over the base ﬁeld k of X , and, moreover, Π U is a characteristic subgroup of Π U S 0 . S 0 Remark 3.5.1. Note that it follows immediately from the deﬁnition of Π Lie U S [cf. also Proposition 3.4, (iii)] that we obtain a natural subgroup def D x Lie = ∗ I x (l) [U S ] ⊗ Q ∗ G k ⊆ Π Lie U S which contains the image of the decomposition group D x ∗ [U S ] ⊆ Π U S via the natural homomorphism Π U S → Π Lie b ≥ 1, D x Lie≤b ⊆ Π Lie≤b U S . Let us write, for Z U S ∗ def def Lie≤b Δ Lie for the image of D x Lie in Π Lie≤b ; I x Lie = D x Lie = D x Lie≤b U S ; I x ∗ U S ∗ ∗ ∗ ∗ [Also, we shall use similar notation when “b” is replaced by “b+”.] Proposition 3.6. Δ Lie≤b . U S (Center-freeness of Lie-iﬁcation) Δ Lie U S is center-free. Proof. Since Δ U S 0 is center-free [cf. Proposition 1.8, (iii)], and the natural morphism Δ Lie U S → Δ U S 0 is surjective, it suﬃces to verify that the centralizer in Lie Lin U S (1/∞) of the image of Δ Lie U S is trivial. But the image of Δ U S in Lin U S (1/∞) contains the image of Δ U S in Lin U S (1/∞). In particular, it follows that the cen- tralizer in question lies in the center of Lin U S (1/∞). Thus, Proposition 3.6 follows from Proposition 3.3, (ii) [or, alternatively, (iii)]. Remark 3.6.1. Observe that changing the choice of splitting G k → D x ∗ [U S ] aﬀects the image of the element F k ∈ G k via the composite of the inclusion G k → Π U S with the morphisms Π U S → Π Lie U S ; Π U S → Π Lie≤b ; U S Π U S → Π Lie≤b+ U S ABSOLUTE ANABELIAN CUSPIDALIZATIONS 59 by conjugation by an element h ∈ I x Lie , which, up to a denominator dividing q k − 1, ∗ lies in the image of I x ∗ [U S ] ⊆ Δ U S — cf. Proposition 3.4, (iv); Proposition 3.6. In particular, it follows that changing the choice of splittings G k → D x ∗ [U S ] aﬀects the Galois invariant splittings of Proposition 3.4, (ii), by conjugation by h. Put another way, if we identify the “Lin U S (1/∞)”, “Lin U S 0 (1/∞)” portions of Δ Lie U S [cf. the deﬁnition of Δ Lie ] with the [l-adic points of the pro-unipotent algebraic U S groups determined by the] corresponding graded Lie objects “Gr Q l (−)(1/∞)” via the Galois invariant splittings of Proposition 3.4, (ii), then it follows that: Changing the choice of splitting G k → D x ∗ [U S ] aﬀects the images of the morphisms Π U S → Π Lie U S ; [where Z Π U S → Π Lie≤b ; U S Π U S → Π Lie≤b+ U S b ≥ 1] by conjugation by h. In light of Proposition 3.6, we may apply the exact sequence “1 → (−) → Aut(−) → Out(−) → 1” [cf. §0] to construct the following topological group: Lie Δ LIE lim U S = ← − Aut(Δ U ) × Out(Δ Lie ) Gal(X k /X k ) def X U S S [where X → X ranges over the (S, S 0 , Σ)-admissible coverings of X; U S ⊆ X is the open subscheme determined by the complement of the set S of closed points of X that lie over points of S]. Note that G k acts naturally on Δ LIE U S ; thus, we may LIE form the semi-direct product of Δ U S with G k to obtain a topological group Π LIE U S . Also, since the various Δ U [where U S ⊆ X is the open subscheme determined by S 0 0 the complement of the set S 0 of closed points of X that lie over points of S 0 ] arising from the X → X that appear in this inverse limit are center-free [cf. Proposition 1.8, (iii)], the natural isomorphism ∼ lim ←− Aut(Δ U S 0 ) × Out(Δ U S ) Gal(X k /X k ) → Δ U S 0 X 0 LIE determines surjections Δ LIE U S Δ U S 0 , Π U S Π U S 0 . Next, let us observe that, for Z b ≥ 1, the various quotients Δ Lie U S Δ tor≤b+1 Δ Lie≤b+ Δ Lie≤b determine quotients of topological groups Δ LIE U S U U U S S S LIE≤b+ TOR≤b+1 Δ U Δ LIE≤b , Π LIE Π LIE≤b+ Π LIE≤b . Δ TOR≤b+1 U S Π U S U S U S U S U S S Thus, we obtain natural homomorphisms of topological groups: TOR≤b+1 LIE≤b+ Δ U Δ LIE≤b Δ U S 0 Δ U S → Δ LIE U S Δ U S U S S TOR≤b+1 LIE≤b+ Π U S → Π LIE Π U Π LIE≤b Π U S 0 U S Π U S U S S We shall denote by LIE≤b+ Δ ≤b+ ; U S ⊆ Δ U S LIE≤b+ Π ≤b+ ; U S ⊆ Π U S LIE≤b Δ ≤b ; U S ⊆ Δ U S LIE≤b Π ≤b U S ⊆ Π U S 60 SHINICHI MOCHIZUKI the respective images of Δ U S , Π U S via these natural homomorphisms. Thus, one ≤b may think of Δ ≤b U S , Π U S as being a sort of “canonical integral structure” on the “inverse limit truncated Lie-iﬁcations” Δ LIE≤b , Π LIE≤b . U S U S Here, we note in passing, relative to the theory of §1, 2, that [it is immediate from the deﬁnitions that] when S = S ∗ [so U S 0 = X], the quotient Π U S Π ≤2 U S is the maximal cuspidally pro-l abelian quotient of Π U S [cf. Proposition 1.14, (i)]. LIE Next, let us observe that in the inverse limit used to deﬁne Δ LIE U S , Π U S , the various “I x Lie ”, “D x Lie ” [cf. Remark 3.5.1] form a compatible system, hence give rise ∗ ∗ to subgroups ⊆ D x LIE ⊆ Π LIE I x LIE U S ; ∗ ∗ I x LIE≤b ⊆ D x LIE≤b ⊆ Π LIE≤b U S ∗ ∗ together with natural exact sequences and isomorphisms [when b ≥ 2] → D x LIE → G k → 1 1 → I x LIE ∗ ∗ → D x LIE≤b → G k → 1 1 → I x LIE≤b ∗ ∗ ∼ ∼ I x LIE = I x LIE≤b = I x ∗ [U S ] ⊗ Q ∗ ∗ (l) [and similarly when “b” is replaced by “b+”]. Also, the images of the subgroups I x ∗ [U S ], D x ∗ [U S ] of Π U S determine subgroups ⊆ D x ≤b ⊆ Π ≤b I x ≤b U S ∗ ∗ [and similarly when “b” is replaced by “b+”]. In the following, let us write [cf. Proposition 3.3, (iv)] def (l) Cnw S ∗ (a/b) = Cnw S ∗ (a/b); Cnw Q S ∗ (a/b) = Cnw S ∗ (a/b) ⊗ Q def (l) [where a, b ∈ Z, 1 ≤ a ≤ b]. Before proceeding, let us observe that [it is immediate from the deﬁnitions that] the natural surjections Δ LIE≤1 Δ U S 0 ; Δ LIE≤1+ U S U S Π LIE≤1+ Π LIE≤1 Π U S 0 U S U S are isomorphisms. On the other hand, for b ≥ 2, we have the following result: Proposition 3.7. Z b ≥ 2: (Plus Liftings of Canonical Integral Structures) For ≤b ≤b+ ≤b (i) The natural surjections Δ ≤b+ U S Δ U S , Π U S Π U S are isomorphisms. LIE≤b (ii) Any two liftings of the natural inclusion Π ≤b to inclusions U S → Π U S ≤b LIE≤b+ LIE≤b+ Π U S → Π U S diﬀer by conjugation in Π U S by a unique element of the LIE≤b+ LIE≤b kernel of Π U S Π U S . ABSOLUTE ANABELIAN CUSPIDALIZATIONS 61 LIE≤b (iii) Any two liftings of the natural inclusion Π ≤b to inclusions U S → Π U S ≤b LIE≤b+ ≤b+ Π U S → Π U S whose images contain D x ∗ in fact coincide. Proof. First, we consider assertion (i). It follows immediately from the deﬁnitions that the kernel in question ≤b ≤b+ ≤b Ker(Δ ≤b+ U S Δ U S ) = Ker(Π U S Π U S ) is contained in [and, in fact, equal to] the inverse limit lim ←− Cnw S ∗ (b + 1/b + 2) X [where X → X ranges over the (S, S 0 , Σ)-admissible coverings of X; S ∗ (respec- tively, S ) is the set of closed points of X that lie over points of S ∗ (respectively, S)]. On the other hand, it follows from the deﬁnition of “Cnw S ∗ (b + 1/b + 2)” that Cnw S ∗ (b + 1/b + 2) is generated by certain successive brackets of the var- (l) ious generators of the Lie algebra Gr(Δ U ) [cf. Proposition 3.3, (i)] with the S property that at least one of the generators appearing in the successive bracket is [in the notation of Proposition 3.3, (i)] either one of the [analogue for X of the] def “α 1 , . . . , α g , β 1 , . . . , β g ” or one of the “ζ x ”, where x ∈ S 0 = S \S ∗ . Moreover, since, by taking Π U ⊆ Π U to be suﬃciently small, one may arrange that the im- S S 0 0 (l) (l) age of Δ U (1/3) in Δ U (1/3) be contained in an arbitrarily small open subgroup S S 0 0 (l) of Δ U (1/3), it thus follows that the above inverse limit vanishes. This completes S 0 the proof of assertion (i). Next, let us observe that to prove assertion (ii), it suﬃces — in light of the natural isomorphism ∼ LIE≤b+ Q Π LIE≤b ) → lim Ker(Π U U S ←− Cnw S (b + 1/b + 2) S X ∗ [where X , S ∗ are as above] — to show that Q H i (Π ≤b U S , Cnw S ∗ (b + 1/b + 2)) = 0 Q for i = 0, 1, each S ∗ as above. Since the action of Δ ≤b U S on Cnw S (b + 1/b + 2) ∗ clearly factors through a ﬁnite quotient of Δ ≤b U S Δ U S 0 , it thus suﬃces to observe [by considering the Leray spectral sequence associated to the surjection Π ≤b U S G k ] Q that the action of F k on Cnw S ∗ (b+1/b+2) is “of weight b+1 ≥ 3”, while the action (l) of F k on (Δ U ) ab is “of weight ≤ 2” [cf. Proposition 3.4, (i)]. This completes the S proof of assertion (ii). Finally, we consider assertion (iii). First, let us observe that any two liftings of LIE≤b LIE≤b+ the natural inclusion Π ≤b to inclusions Π ≤b whose images U S → Π U S U S → Π U S 62 SHINICHI MOCHIZUKI ∼ contain D x ≤b+ → D x ≤b [since b ≥ 2] in fact coincide on D x ≤b ⊆ Π ≤b U S . Thus, by ∗ ∗ ∗ assertion (ii), it suﬃces to verify that the submodule of F k -invariants of LIE≤b+ Π LIE≤b ) Ker(Π U U S S is zero. But in light of the natural isomorphism ∼ LIE≤b+ Q Π LIE≤b ) → lim Ker(Π U U S ←− Cnw S (b + 1/b + 2) S ∗ X [where X , S ∗ are as above], this follows from Proposition 3.4, (i). This completes the proof of assertion (iii). Next, for Z b ≥ 1, let us denote by ⊆ Δ TOR≤b+1 ; Δ ≤b++ U S U S Π ≤b++ ⊆ Π TOR≤b+1 U S U S the respective images of Δ U S , Π U S via the natural homomorphisms considered above and by ⊆ D x ≤b++ ⊆ Π ≤b++ I x ≤b++ U S ∗ ∗ the images of the subgroups I x ∗ [U S ], D x ∗ [U S ] of Π U S . Observe that it follows from , Π TOR≤b+1 [cf. also Proposition 3.3, (iv)] that the the deﬁnition of Δ TOR≤b+1 U S U S ≤b++ ≤b+ ≤b++ natural surjections Δ U S Δ U S , Π U S Π ≤b+ are, in fact, isomorphisms. U S Thus, by Proposition 3.7, (i), we obtain a commutative diagram of natural homo- morphisms Π ≤b+1 U S ⏐ ⏐ LIE≤b+1 Π U S Π ≤b++ U S ⏐ ⏐ Π TOR≤b+1 U S → ∼ Π ≤b+ U S ⏐ ⏐ → ∼ Π ≤b U ⏐ S ⏐ Π LIE≤b+ U S Π LIE≤b U S [where the vertical arrows are the natural inclusions; all of the horizontal arrows are surjections; the second two upper horizontal arrows are isomorphisms]. Moreover, it follows immediately from the deﬁnitions that the ﬁrst square in this commuta- ⊆ Π LIE≤b+1 may be tive diagram is cartesian. That is to say, the subgroup Π ≤b+1 U S U S LIE≤b+1 TOR≤b+1 thought of as the inverse image via the natural surjection Π U S Π U S ≤b LIE≤b+ of the image of a certain lifting of the natural inclusion Π U S → Π U S [cf. Propo- ≤b TOR≤b+1 sition 3.7, (i)] to an inclusion Π U S → Π U S . Also, let us write: ≤b Π ≤b U S [csp] = Ker(Π U S Π X ) def [csp] = Ker(Π ≤b++ Π X ) Π ≤b++ U S U S def ≤b++ for the cuspidal subgroups of Π ≤b . U S , Π U S ABSOLUTE ANABELIAN CUSPIDALIZATIONS 63 Next, following the pattern of §1, we relate the constructions made so far to the fundamental groups Δ U X×X , Π U X×X [cf. the discussion preceding Proposition 1.6]. For simplicity, we assume from now on that: S = S ∗ = {x ∗ } [so S 0 = ∅]. Write D x ∗ [X] ⊆ Π X for the image of D x ∗ [U S ] via the natural surjection Π U S Π X . Then the projection Π U X×X Π X to the second factor determines a natural isomorphism ∼ Π U S → Π U X×X × Π X D x ∗ [X] [cf. Proposition 1.8, (ii)]. Moreover, this isomorphism determines a natural iso- morphism ∼ (Π U S ⊇) D x ∗ [U S ] → D X × Π X D x ∗ [X] (⊆ D X ⊆ Π U X×X ) [where “D X ” is as in the discussion preceding Proposition 1.12] which is compat- ible with the natural inclusions D x ∗ [U S ] → Π U S , D X → Π U X×X . Put another way, D x ∗ [U S ] [hence also I x ∗ [U S ], G k ⊆ D x ∗ [U S ]] may be thought of as being “si- multaneously” a subgroup of both Π U S and D X . Thus, we obtain a natural exact sequence 1 −→ Δ U S −→ Π U X×X −→ Π X −→ 1 together with compatible inclusions Δ U S ⊇ Δ U ⊇ I x ∗ [U S ] ⊆ D x ∗ [U S ] ⊆ D X ⊆ Π U X×X S [where X → X is an (S, ∅, Σ)-admissible covering of X; U S ⊆ X is the open subscheme determined by the complement of the set S of closed points of X that lie over x ∗ ]. Also, we shall write: def Δ D X = D X Δ U X×X ⊆ Π U X×X In particular, we obtain natural actions [by conjugation] of D X on Δ U S , Δ U S [as well as on the various objects naturally constructed from Δ U S , Δ U in the above S discussion], which we shall refer to as diagonal actions. Proposition 3.8. (Characterization of the Diagonal Action) Suppose that S = S ∗ = {x ∗ }. Then in the notation and terminology of the above discussion, the diagonal action of D X on Lin U S (1/∞) is completely determined [i.e., as a continuous action of the topological group D X on the topological group Lin U S (1/∞)] by the following conditions: (a) the action is compatible with the natural action of D x ∗ [U S ] ⊆ D X on Lin U S (1/∞); (b) the action is compatible with the ﬁltration {Lin U S (a/∞)} [where a ≥ 1 is an integer] on Lin U S (1/∞). 64 SHINICHI MOCHIZUKI (c) the action coincides with the diagonal action of D X on the quotient Lin U S (1/4) [cf. condition (b)] of Lin U S (1/∞). Proof. First, I claim that it suﬃces to show that these conditions determine the def Δ Δ Δ action of the subgroup D X /X = D X × Δ X Δ X ⊆ D X ⊆ D X on Lin U (1/∞). S Δ is determined, it follows that the action of Indeed, once the action of D X /X def Δ D X /X = D X × Π X Π X ⊆ D x ∗ [U S ] · D X /X ⊆ D X is determined [cf. condition (a)]. On the other hand, since Π X is an open normal subgroup of Π X , it follows that D X /X is an open normal subgroup of D X . Thus, by considering the conjugation actions of D X on D X /X and of Im(D X ) ⊆ Lin U (1/∞) S on Im(D X /X ) ⊆ Lin U (1/∞) [i.e., of the group of automorphisms of Lin U (1/∞) S S induced by elements of D X on the group of automorphisms of Lin U (1/∞) induced S by elements of D X /X ], we conclude that the action of D X on Lin U S (1/∞) is de- termined up to composition with automorphisms of Lin U S (1/∞) that commute with the action of D X /X and [cf. condition (c)] induce the identity on the quo- tient Lin U S (1/4). Now let α be an automorphism of Lin U S (1/∞) that commutes with the action of D X /X and induces the identity on the quotient Lin U S (1/4). Then α commutes with some open subgroup of G k ⊆ D x ∗ [U S ] ⊆ D X , so α induces an automorphism of Lie U S (1/∞) that is compatible with the splittings of Propo- (l) sition 3.4, (ii). Since Gr(Δ U ) is generated by its elements “of weight ≤ 2” [cf. S Proposition 3.3, (i)], we thus conclude that α induces the identity automorphism of Lie U (1/∞), hence that α itself is the identity automorphism. This completes S the proof of the claim. Next, let us observe that by condition (c) [cf. also Proposition 3.3, (i)], the Δ action of D X /X on Lin U (1/∞) is unipotent, relative to the ﬁltration of condition S (b). Thus, it follows [from the deﬁnition of “Lie(−)”] that the induced action of Δ D X /X on Lie U (1/∞) determines an action of the Lie algebra S def Δ Δ (l) (1/∞)) Lie(D X /X ) = Lie((D X /X ) Δ (l) Δ (l) for the maximal pro-l quotient of (D X ] on the Lie [where we write (D X /X ) /X ) algebra Lie U (1/∞). Moreover, to complete the proof of Proposition 3.8, it suﬃces S to show that this Lie algebra action is the action arising from the diagonal action. In fact, since this Lie algebra action is compatible [cf. condition (a)] with the actions of Δ G k on Lie(D X /X ), Lie U (1/∞), it follows, by considering the induced eigenspace S splittings [cf. Proposition 3.4, (ii)], that [to complete the proof of Proposition 3.8] def Δ Δ it suﬃces to show that the Lie algebra action of Gr(D X /X ) = Gr(Lie(D X /X )) on (l) Gr(Δ U ) is the action arising from the diagonal action. On the other hand, since S (l) Δ Gr(D X /X ), Gr(Δ U ) are generated by elements “of weight ≤ 2” [cf. S Proposition ABSOLUTE ANABELIAN CUSPIDALIZATIONS 65 3.3, (i)], this follows by observing that the Lie algebra action of the unique generator Δ Δ of Gr(D X /X ) “of weight 2” [which arises from I x ∗ [U S ] ⊆ D X /X ] is determined by Δ condition (a), while the Lie algebra action of the generators of Gr(D X /X ) “of weight (l) (l) S S 1” [which send elements of Gr(Δ U ) “of weight ≤ 2” to elements of Gr(Δ U ) “of weight ≤ 3”] is determined by condition (c). This completes the proof of Proposition 3.8. Remark 3.8.1. Note that the conditions of Proposition 3.8 allow one to char- acterize not only the diagonal action of D X on Lin U S (1/∞), but also on Δ Lie U , S LIE Π Lie , Π LIE , hence also on Δ U U S [where we note that the diagonal action of D X on U S S Gal(X k /X k ) is simply the conjugation action arising from the quotients D X Π X , Δ X Gal(X k /X k )]. Remark 3.8.2. Note that the groups Lin U S (1/4) of condition (c) of Proposition 3.8 are, as groups equipped with the surjection Lin U S (1/4) Lin X (1/4), cuspi- dally abelian [i.e., the kernel of this surjection is abelian], hence may be constructed from the maximal cuspidally abelian quotients Π U X×X Π c-ab U X×X of Theorem 1.16. Proposition 3.9. (Extensions of Canonical Integral Structures) Suppose that S = S ∗ = {x ∗ } [cf. Remark 3.9.2 below]. Let b ≥ 1 be an integer. Then: (i) Suppose that b = 1. Then any two liftings of the natural inclusion Π ≤b U S → LIE≤b+ ≤b TOR≤b+1 TOR≤b+1 Π U S to inclusions Π U S → Π U S diﬀer by conjugation in Π U S by TOR≤b+1 LIE≤b+ an element of the kernel of Π U S Π U S . (ii) Suppose that b ≥ 2. Then any two liftings of the natural inclusion Π ≤b U S → LIE≤b+ ≤b TOR≤b+1 Π U S to inclusions Π U S → Π U S whose images contain I x ≤b++ diﬀer by ∗ TOR≤b+1 TOR≤b+1 LIE≤b+ conjugation in Π U S by an element of the kernel of Π U S Π U . S that satisﬁes the (iii) Let β be an automorphism of the proﬁnite group Π ≤b+1 U S following two conditions: (a) β preserves and induces the identity on the quo- ≤b+1 tient Π ≤b+1 Π ≤b ⊆ Π ≤b+1 . Then β is a U S U S ; (b) β preserves the subgroup I x ∗ U S ≤b+1 ≤b Ker(Π U S Π U S )-inner automorphism. LIE≤b+ Π U ) be an element that is invariant (iv) Let δ ∈ Ker(Π TOR≤b+1 U S S under the diagonal action of D X . Then if b = 1, then δ lies in the image of I x ∗ [U S ] ⊗ (Q l /Z l ); if b ≥ 2, then δ is the identity element. (v) Write ≤b lim Π U S Π ≤∞ U S = ← − Π U S ; def b ≤b Δ U S Δ ≤∞ lim U S = ← − Δ U S def b 66 SHINICHI MOCHIZUKI ≤b for the the quotients of Π U S , Δ U S deﬁned by the inverse limit of the Π ≤b U S , Δ U S and Π U X×X Π ≤∞ U X×X ; Δ U X×X Δ ≤∞ U X×X for the quotients of Π U X×X , Δ U X×X determined by the kernel in Δ U S ⊆ Δ U X×X ⊆ Π U X×X [cf. the discussion preceding Proposition 3.8] of Ker(Δ U S Δ ≤∞ U S ). Then ≤∞ ≤∞ ≤∞ Π U S Π U S (respectively, Δ U S Δ U S ; Π U X×X Π U X×X ; Δ U X×X Δ ≤∞ U X×X ) is the maximal cuspidally pro-l quotient of Π U S (respectively, Δ U S ; Π U X×X ; ≤∞ ≤∞ ≤∞ † † Δ U X×X ); moreover, (Π ≤∞ U S ) , Δ U S , (Π U X×X ) , Δ U X×X [where the daggers denote the result of applying the operation “× G k G † k ”] are slim. Proof. First, we consider assertions (i), (ii). Observe that, for Z b ≥ 1, the ≤b LIE≤b+ diﬀerence of any two liftings of the natural inclusion Π U S → Π U S to inclusions ≤b TOR≤b+1 Π U S → Π U S determines a compatible collection of cohomology classes tor η S ∈ H 1 (Π ≤b U S , New S (b + 1/b + 2)) [where X → X ranges over the (S, ∅, Σ)-admissible coverings of X; S is the set of closed points of X that lie over x ∗ ]. Since New tor S (b + 1/b + 2) = 0 whenever b is even, we may assume for the remainder of the proof of assertions (i), (ii) that b is odd. Next, let us observe that by Proposition 3.4, (i), the zeroth cohomology module tor H 0 (Π ≤b U S , New S (b + 1/b + 2)) is ﬁnite. This ﬁniteness implies that any [not necessarily compatible!] system of tor sections of a compatible system of torsors over H 0 (Π ≤b U S , New S (b + 1/b + 2)) always admits a compatible coﬁnal subsystem. In light of the natural isomorphism ∼ LIE≤b+ tor Π U ) → lim Ker(Π TOR≤b+1 U S ←− New S (b + 1/b + 2)) S X [where X , S are as described above], we thus conclude that in order to show TOR≤b+1 that the two inclusions Π ≤b diﬀer by conjugation by an element of U S → Π U S TOR≤b+1 LIE≤b+ Ker(Π U S Π U S ), it suﬃces to show that the η S = 0. tor Note that Π ≤b U S [csp] acts trivially on New S (b + 1/b + 2)). Now I claim that: Each η S arises from a unique class [which, by abuse of notation, we shall also denote by η S ] in H 1 (Π X , New tor S (b + 1/b + 2)). Indeed, if b = 1, this claim follows from the fact that Π ≤b U S [csp] = {1} [cf. the discus- sion preceding Proposition 3.7], so assume that b ≥ 2, and that we are in the situa- tion of assertion (ii). Now observe that since S = S ∗ is of cardinality one, it follows ABSOLUTE ANABELIAN CUSPIDALIZATIONS 67 ≤b++ that Π ≤b [csp]) is topologically generated by the Π ≤b U S [csp] (respectively, Π U S U S - (re- ≤b++ ≤b ≤b++ spectively, Π U S -) conjugates of I x ∗ (respectively, I x ∗ ). Note, moreover, that it LIE≤b+ is immediate from the deﬁnitions that every element of Ker(Π TOR≤b+1 Π U ) U S S ≤b ≤b++ . In particular, it follows that the images of Π U S [csp] via the commutes with I x ∗ ≤b two inclusions Π U S → Π TOR≤b+1 under consideration necessarily coincide. But this U S implies that each η S arises from a unique class in H 1 (Π X , New tor S (b + 1/b + 2)), thus completing the proof of the claim. Next, [returning to the general situation involving both assertions (i) and (ii)] let X → X be a morphism of (S, ∅, Σ)-admissible coverings of X. Write U S ⊆ X for the open subscheme determined by the complement of the set S of closed points of X that lie over points of S. Also, let us assume that the open subgroup Δ X ⊆ Δ X arises from some open subgroup H ⊆ Δ ab X that is preserved by the action of Π X . Thus, it def follows that the covering X k → X k is abelian; write Gal(X /X ) = Gal(X k /X k ). For c a positive integer, set: def R = Z l ; def def R c = Z l [c · Gal(X /X )] ⊆ R = Z l [Gal(X /X )] [where we write c·Gal(X /X ) ⊆ Gal(X /X ) for the subgroup of the abelian group Gal(X /X ) that arises as the image of multiplication by c]. Thus, R (respectively, R ; R c ) is a commutative ring with unity whose underlying R - (respectively, R c -; R -) module is ﬁnite and free; moreover, R , R c admit a natural Π X -action [induced by the conjugation action of Π X on the subquotient Gal(X /X ) of Π X ]. Also, we shall denote by c : R c R ; : R R the augmentations obtained by mapping all of the elements of Gal(X /X ) to 1. Next, let us observe that S , S admit natural Π X -actions with respect to which we have natural isomorphisms of Π X -modules [cf. Proposition 3.3, (i), (iv)] ∼ New S (2/3) → R [S ] ⊗ M X ; (l) ∼ New S (2/3) → R [S ] ⊗ M X (l) which determine natural isomorphisms of Π X -modules ∼ (l) ∼ (l) New S (2c/2c + 1) → Lie cR (R [S ] ⊗ M X ) New S (2c/2c + 1) → Lie cR (R [S ] ⊗ M X ) [cf. the notation of Proposition A.1] for integers c ≥ 1. In the following, we shall identify the domains and codomains of these isomorphisms via these isomorphisms. Next, let us observe that the R -module R [S ] admits a natural R -module structure that is compatible with the Π X -action on R , R [S ]. Note, moreover, that R [S ] is a free R -module, and that we have a natural isomorphism ∼ R [S ] ⊗ R , R → R [S ] 68 SHINICHI MOCHIZUKI induced by the augmentation : R R . Also, we observe that any choice of rep- resentatives in S of the Δ X /Δ X = Gal(X /X )-orbits of S [where we note that the set of such orbits may be naturally identiﬁed with S ] determines an R -basis of R [S ], hence [by considering “Hall bases” — cf., e.g., [Bour], Chapter II, §2.11] an R -basis of Lie cR (R [S ]). Note that since the natural action of Gal(X /X ) on Lie cR (R [S ]) is compatible with the Lie algebra structure, it follows that: This natural action of Gal(X /X ) on Lie cR (R [S ]) is given by composing the R -module structure action Gal(X /X ) → R with the morphism c· : Gal(X /X ) → Gal(X /X ) given by multiplication by c. In particular, this natural action of Gal(X /X ) on Lie cR (R [S ]) factors through the quotient Gal(X /X ) c·Gal(X /X ) and hence determines on Lie cR (R [S ]) a structure of “induced” c· Gal(X /X )-module [in the terminology of the cohomol- ogy theory of ﬁnite groups]. Thus, we obtain natural, Π X -equivariant isomorphisms ∼ R [S ] → R [S ] Gal(X /X ) ∼ Lie cR (R [S ]) ⊗ R c , c R → Lie cR (R [S ]) Gal(X /X ) = Lie cR (R [S ]) c·Gal(X /X ) [where we use superscripts to denote the submodules of invariants with respect to the action of the superscripted group]. Moreover, we observe that relative to these natural isomorphisms, the restriction of the natural surjection Lie cR (R [S ]) Lie cR (R [S ]) ⊗ R c , c R to the submodule of Gal(X /X )-invariants induces the endomorphism of the module Lie cR (R [S ]) ⊗ R c , c R given by multiplication by the order of c · Gal(X /X ). Now let us write: def c New tor S /S (2c/2c + 1) = Lie R (R [S ] ⊗ M X ) ⊗ (Q l /Z l ) (l) def tor R New tor S /S (2c/2c + 1) = New S /S (2c/2c + 1) ⊗ R c , c [where c ≥ 1 is an integer]. Then in light of the above observations [together with Propositions A.1, (iv); 3.3, (iv)], we conclude the following: (A) The natural surjection of Π X -modules tor New tor S (b + 1/b + 2) New S (b + 1/b + 2) admits a factorization tor tor New tor S (b + 1/b + 2) New S /S (b + 1/b + 2) New S /S (b + 1/b + 2) New tor S (b + 1/b + 2) [via morphisms of Π X -modules]. Moreover, the natural action of Δ X on the module New tor S /S (b + 1/b + 2) factors through the quotient Δ X Gal(X /X ) c · Gal(X /X ) and determines on New tor S /S (b + 1/b + 2) a structure of induced c · Gal(X /X )-module. ABSOLUTE ANABELIAN CUSPIDALIZATIONS 69 (B) The induced morphism on Δ X -invariants Δ X Δ X New tor → New tor = New tor S (b + 1/b + 2) S (b + 1/b + 2) S (b + 1/b + 2) of the [ﬁrst] natural surjection of (A) factors, in a Π X -equivariant fashion, through the endomorphism tor New tor S /S (b + 1/b + 2) → New S /S (b + 1/b + 2) tor [hence also through the endomorphism New tor S (b + 1/b + 2) → New S (b + 1/b + 2)] given by multiplication by the order of c · Gal(X /X ). Also, before proceeding, we make the following elementary observation con- cerning the group cohomology of induced modules: ab (C) Suppose that H = l n · Δ ab X ⊆ Δ X , where n is a positive integer. For M a ﬁnitely generated Z l -module [which we regard as equipped with the trivial Δ X -action], write: def (l) H X = H 1 (Δ X , M ⊗ M X ) def ∼ H X = H 1 (Δ X , M ⊗ M X ) → H 1 (Δ X , M [Gal(X /X )] ⊗ M X ) (l) (l) Then the “trace map” Tr H : H X → H X — i.e., the map induced by the morphism of coeﬃcients M [Gal(X /X )] M that maps each element of Gal(X /X ) to 1 — factors through the endomorphism of H X given by multiplication by l n [cf. Remark 3.9.2 below]. [Indeed, to verify (C), we recall that this trace map Tr H is well-known to be dual [via Poincaré duality — cf., e.g., [FK], pp. 135-136] to the pull-back morphism; ∼ we thus conclude that, relative to the natural isomorphisms H X → Δ ab X ⊗ M , ∼ ab H X → Δ X ⊗ M [arising from Poincaré duality — cf., e.g., Proposition 1.3, (ii)], the trace map corresponds to the natural morphism ab H X = Δ ab X ⊗ M → Δ X ⊗ M = H X induced by the inclusion Δ X ⊆ Δ X — hence, by the deﬁnition of H , factors through the endomorphism of H X given by multiplication by l n . This completes the proof of (C).] Next, let us suppose that we have been given morphisms of (S, ∅, Σ)-admissible coverings of X ∗ ∗ → X → X → X X → X ∗ ∗ ∗ ∗ and write U S ⊆ X , U S ∗ ⊆ X , U S ∗ ⊆ X for the open subscheme deter- ∗ ∗ mined, respectively, by the complements of the sets S , S , S of closed points 70 SHINICHI MOCHIZUKI ∗ ∗ of X , X , X that lie over points of S. Also, let us assume that the open sub- groups Δ X ⊆ Δ X , Δ X ∗ ⊆ Δ X , Δ X ⊆ Δ X , Δ X ∗ ⊆ Δ X arise, respectively, from open subgroups H = l n · Δ ab ⊆ H X ∗ ∗ ∗ H = l n · Δ ab X ⊆ H ∗ ∗ def ∗ = l n ∗ = l n · Δ ab ⊆ Δ ab X ∗ X ∗ ab · Δ ab X ⊆ Δ X def — where n = n − c, n = n − c; we suppose that n > 2c, n > c are “suﬃciently large” positive integers, to be chosen below. Then we wish to apply the theory developed above [in particular, the observations (A), (B), (C)] by taking “X → X ” in this theory to be various subcoverings of X → X . Now let us compute the cohomology of Π X via the Leray spectral sequence associated to the surjection Π X Π X /Δ X ∗ . Suppose that c has been chosen so ∗ that b + 1 = 2c. Then by applying (A) to the covering “X → X ” (respectively, ∗ “X → X ”), we conclude that Δ X ∗ (respectively, Δ X ∗ ) acts trivially on tor New tor S /S ∗ (b + 1/b + 2) (respectively, New S /S ∗ (b + 1/b + 2)). Also, it follows immediately from the deﬁnitions that we have a natural Π X -equivariant surjection tor New tor S /S ∗ (b + 1/b + 2) New S /S ∗ (b + 1/b + 2). Now, by applying (A) to the ∗ covering “X → X ” and (C) to the covering “X the Π X -equivariant natural morphism ∗ → X ∗ ”, we conclude that tor 1 H 1 (Δ X ∗ , New tor ∗ (b + 1/b + 2)) → H (Δ ∗ , New ∗ (b + 1/b + 2)) X S /S S /S [which maps the image of η S to the image of η S !] factors through a “trace ∗ ∗ → X ”, hence in particular, through the map” as in (C) for the covering “X endomorphism of H 1 (Δ X ∗ , New tor ∗ (b + 1/b + 2)) [a module whose submodule S /S ∗ of Π X -invariants is ﬁnite, by Proposition 3.4, (i)] given by multiplication by n , in a Π X -equivariant fashion. Thus, by taking n to be “suﬃciently large”, we conclude that the image of η S in H 1 (Δ X ∗ , New tor (b + 1/b + 2)) is zero. S /S ∗ Now I claim that the image of η S in H 1 (Δ X , New tor S /S (b + 1/b + 2)) [obtained by applying the surjection tor New tor S (b + 1/b + 2)) New S /S (b + 1/b + 2) of (A) applied to the covering “X → X ”] is zero. Indeed, note that it follows im- mediately from the deﬁnitions that we have a natural surjection New tor ∗ (b+1/b+ S /S ∗ 2) New tor S /S (b + 1/b + 2) [induced, in eﬀect, by the inclusion Gal(X /X ) → Gal(X /X )]. Thus, since we have already shown that the image of η S in the co- homology module H 1 (Δ X ∗ , New tor ∗ (b+1/b+2)) is zero, it follows immediately S /S 1 that the image of η S in H (Δ X ∗ , New tor S /S (b + 1/b + 2)) is zero, hence that the image in question in the claim arises from a class ∈ H 1 (Gal(X ∗ Δ X ∗ /X ), (New tor ) S /S (b + 1/b + 2)) = H 1 (Gal(X ∗ /X ), New tor S /S (b + 1/b + 2)) = 0 ABSOLUTE ANABELIAN CUSPIDALIZATIONS 71 [where the last cohomology module vanishes since, by (A) applied to the covering ∗ “X → X ”, New tor S /S (b + 1/b + 2) is an induced Gal(X /X )-module]. This completes the proof of the claim. Thus, in summary, we conclude that the image of η S in the cohomology module H 1 (Π X , New tor S /S (b + 1/b + 2)) arises from a unique class in ∼ Δ X H 1 (Π X /Δ X , (New tor ) → H 1 (Π X /Δ X , New tor S /S (b + 1/b + 2)) S /S (b + 1/b + 2)) which maps to the unique class in H 1 (Π X /Δ X , New tor S (b + 1/b + 2)) [a module which is ﬁnite, by Proposition 3.4, (i)] that gives rise to η S via a mor- phism that factors through the endomorphism given by multiplication by the order of c · Gal(X /X ) [cf. (A), (B) applied to the covering “X → X ”]. In particular, by taking n to be “suﬃciently large”, we may conclude that η S = 0, as desired. That is to say: TOR≤b+1 This completes the proof that the two inclusions Π ≤b diﬀer U S → Π U S TOR≤b+1 LIE≤b+ by conjugation by an element of Ker(Π U S Π U S ). In particular, the proof of assertions (i), (ii) is complete. Next, we consider assertion (iii). First, let us observe that when b = 1, assertion (iii) follows immediately from [the “pro-l version” of the argument applied to prove] Proposition 2.6, (i) [cf. the discussion preceding Proposition 3.7]. Thus, in the remainder of the proof of assertion (iii), we assume that b ≥ 2. Note that since the ≤b+1 Π ≤b , elements of Ker(Π ≤b+1 U S U S ) manifestly commute with the elements of I x ∗ it follows from conditions (a), (b), the fact that b ≥ 2, and the assumption that [csp] [cf. the proof S = S ∗ is of cardinality one that β induces the identity on Π ≤b+1 U S of assertion (ii) above]. Thus, to complete the proof of assertion (iii), it suﬃces to show that the compatible system of classes λ S ∈ H 1 (Π X , New S (b + 1/b + 2)) determined by β [cf. Proposition 3.7, (i); 3.3, (iv)] vanishes. Note that since (l) (Δ X ) ab is of “weight ≤ 1”, and New S (b + 1/b + 2) is of “weight b + 1 ≥ 3” [cf. Proposition 3.4, (i)], it follows immediately from the Leray spectral sequence for Π X G k that we have a natural isomorphism ∼ H 1 (G k , (New S (b + 1/b + 2)) Δ X ) → H 1 (Π X , New S (b + 1/b + 2)) [where the superscript “Δ X ” denotes the Δ X -invariants] and that the module H 1 (G k , (New S (b + 1/b + 2)) Δ X ) is ﬁnite. Thus, to show that the λ S = 0, it suﬃces to show that the inverse limit Δ X lim ←− (New S (b + 1/b + 2)) X 72 SHINICHI MOCHIZUKI [where X , S are as described in the proof of assertions (i), (ii)] is zero. But this follows from observation (B) of the proof of assertions (i), (ii). This completes the proof of assertion (iii). , it suf- Next, we consider assertion (iv). In light of the deﬁnition of Π TOR≤b+1 U S ﬁces to show that any compatible system of D X -invariant [relative to the diagonal action of D X ] classes κ S ∈ New tor S (b + 1/b + 2) [where X , S are as described in the proof of assertions (i), (ii)] lies in the image of I x ∗ [U S ] ⊗ (Q l /Z l ) if b = 1 and vanishes if b ≥ 2. [Here, we note that since New tor S (b + 1/b + 2) = 0 when b is even, we may assume without loss of generality that b is odd.] To do this, let X , X , S , S be as in (A), (B). Now we would like to apply the theory of the Appendix [cf., especially, Theorem A.5] to the present situation. To do this, it is necessary to specify the data “(i), (ii), (iii), (vi), (vii), (viii), (ix), (x), (xi), (xii)” [cf. the discussion of the Appendix] to which this theory is to be applied. We take the “d” of Theorem A.5 to be such that 2d = b + 1 [so the fact that b is odd implies that d ≥ 2 whenever b ≥ 2] and the prime number “l” of “(i)” to be the prime number l of the present discussion. We take the proﬁnite group “Δ” of “(ii)” to be the quotient of the group Δ X by the kernel of the quotient ab (Δ X ⊇) Δ X Δ ab X Δ X ⊗ Z l ; this group “Δ” surjects onto Δ X /Δ X , which we take to be the quotient group “G” of “(ii)”, with kernel Δ ab X ⊗ Z l , which we take to be the subgroup “V ” of “(ii)”. Here, we recall that the condition of “(ii), (c)” concerning the regular representation follows immediately from [Milne], p. 187, Corollary 2.8 [cf. also [Milne], p. 187, Remark 2.9], in light of our assumption that X is proper hyperbolic, hence of genus ≥ 2. We take the proﬁnite group “Γ” of “(ix)” to be the image G k ⊆ G k of Π X in G k [so “Γ” acts naturally on “Δ”, “G”, “H”]. Thus, “Δ Γ ” may be thought of as a quotient of Π X × G k G k , hence also as a quotient of D X × G k G k . Note that by consideration of “weights”, it follows that G k (New tor S (b + 1/b + 2)) is ﬁnite, hence annihilated by some ﬁnite power of l, which we take to be the number “N ” of “(iii)”. We take the covering X → X of (A), (B) to be any (S, ∅, Σ)- admissible covering such that the resulting covering X k → X k is the covering determined by the resulting subgroup “l n · V ⊆ Δ” [cf. the statement of Theorem A.5], so “J” may be identiﬁed with Gal(X /X ). Next, we take the “G-torsor E G ” of “(vi)” to be S and the “H-torsor E H ” of “(vii)” to be S ; thus, the natural surjection S S determines a surjection “E H E G ” as in “(viii)”. Note that S (respectively, S ) may be thought of as a Δ U S -orbit (respectively, Δ U S - orbit) [via the action by conjugation] of the conjugacy class of subgroups of Δ U S determined by I x ∗ [U S ] ⊆ Δ U S . In particular, it follows that the particular member of this conjugacy class constituted by the subgroup I x ∗ [U S ] ⊆ Δ U S determines a particular element e H ∈ E H (respectively, e G ∈ E G ) as in “(xi)”. Moreover, the diagonal action of D X — hence also of D X × G k G k ⊆ D X — on Δ U S determines an action of D X × G k G k ⊆ D X on E H , E G that ﬁxes e H , e G , and [as is easily veriﬁed] ABSOLUTE ANABELIAN CUSPIDALIZATIONS 73 factors through the quotient “Δ Γ ” of D X × G k G k Π X × G k G k ; in particular, we obtain continuous actions of “Δ Γ ” on “E G ”, “E H ” as in “(x)”. Finally, we take (l) the “Γ-module Λ” of “(xii)” to be the d-th tensor power of M X ⊗ (Q l /Z l ). This completes the speciﬁcation of the data necessary to apply Theorem A.5. Thus, by applying Theorem A.5 to the composite of the second and third surjections in the factorization of (A), we conclude that since κ S is D X -invariant, it follows that κ S ∈ New tor S (b + 1/b + 2) G k = 0 when b ≥ 2 and to maps to an element [i.e., κ S ] of N · New tor S (b + 1/b + 2) an element [i.e., κ S ] in the image of I x ∗ [U S ] ⊗ (Q l /Z l ) when b = 1. This completes the proof of assertion (iv). Finally, we consider assertion (v). It is immediate from the deﬁnitions that the various quotients in question are cuspidally pro-l. That these quotients are the maximal cuspidally pro-l quotients follows from the construction of Δ ≤∞ U S and the (l) (l) easily veriﬁed fact that each Δ U injects into Lin(Δ U (1/∞))(Q l ). Finally, the S S asserted slimness follows from the fact that the proﬁnite groups in question may be written as inverse limits of proﬁnite groups that admit normal open subgroups (l) (l) (l) (l) [with trivial centralizers] — namely, “Δ U ”, “(Π U ) † ”, “Δ U X ×X ”, “(Π U X ×X ) † ” S S — which are slim, by Proposition 1.8, (i), (iii) [which implies that the quotients (l) (l) (l) (l) Δ U X ×X Δ X , (Π U X ×X ) † (Π X ) † , as well as the kernels of these quotients, are slim]. Remark 3.9.1. Proposition 3.9, (iii), may be regarded as a “higher order, pro-l analogue” of Proposition 2.6, (i). Remark 3.9.2. It is important to note that if one omits [as was, mistakenly, done in an earlier version of this paper] the hypothesis that S 0 = ∅, then it no longer holds that the image of the trace map “Tr H : H X → H X ” [appearing in the proof of Proposition 3.9, (i), (ii)] lies in l n · H X . Indeed, this phenomenon may be understood by considering the trace map on ﬁrst étale cohomology modules with Z l -coeﬃcients associated to the l n -th power map G m → G m on the multiplicative group G m over k — a map which, as an easy computation reveals, is surjective. We are now ready to prove the main technical result of the present §3: Theorem 3.10. (Reconstruction of Maximal Cuspidally Pro-l Exten- sions) Let X, Y be proper hyperbolic curves over a ﬁnite ﬁeld; denote the base ﬁelds of X, Y by k X , k Y , respectively. Suppose further that we have been def def def given points x ∗ ∈ X(k X ), y ∗ ∈ Y (k Y ); write S = {x ∗ }, T = {y ∗ } U S = X\S, def V T = Y \T . Let Σ be a set of prime numbers that contains at least one prime number that is invertible in k X , k Y ; thus, Σ determines various quotients Π U S , 74 SHINICHI MOCHIZUKI Π X , Π U X×X , Π X×X , Π V T , Π Y , Π U Y ×Y , Π Y ×Y [cf. Proposition 1.8, (iii); the dis- cussion preceding Proposition 1.6] of the étale fundamental groups of U S , X, U X×X , X × X, V T , Y , U Y ×Y , Y × Y , respectively. Also, we write Π X G k X , Π Y G k Y for the quotients determined by the respective absolute Galois groups of k X , k Y . Let ∼ α : Π X → Π Y be a Frobenius-preserving [hence also quasi-point-theoretic — cf. Remark 1.18.2] isomorphism of proﬁnite groups that maps the decomposition group of x ∗ in Π X [which is well-deﬁned up to conjugation] to the decomposition group of y ∗ in Π Y [which is well-deﬁned up to conjugation]. Then for each prime l ∈ Σ such that l = p, there exist commutative diagrams Π ≤∞ U S ⏐ ⏐ ∞ −→ Π X −→ α α Π ≤∞ V T ⏐ ⏐ Π ≤∞ U X×X ⏐ ⏐ Π Y Π X×X α × ∞ −→ α×α −→ Π ≤∞ U Y ×Y ⏐ ⏐ Π Y ×Y ≤∞ ≤∞ ≤∞ — in which Π U S Π ≤∞ U S , Π U X×X Π U X×X , Π V T Π V T , Π U Y ×Y Π U Y ×Y are the maximal cuspidally pro-l quotients [cf. Proposition 3.9, (v)]; Π X×X ∼ = Π × Π ; the vertical arrows are the natural surjections; Π X × G kX Π X , Π Y ×Y ∼ = Y G kY Y are isomorphisms, well-deﬁned up to composition with a cuspidally inner α ∞ , α × ∞ automorphism, that are compatible, relative to the natural surjections c-ab,l Π ≤∞ U S Π U S ; c-ab,l Π ≤∞ U X×X Π U X×X ; c-ab,l Π ≤∞ ; V T Π V T c-ab,l Π ≤∞ U Y ×Y Π U Y ×Y — where we use the superscript “c-ab, l” to denote the respective maximal cusp- idally pro-l abelian quotients — with the isomorphisms ∼ c-ab Π c-ab U S → Π V T ; ∼ c-ab Π c-ab U X×X → Π U Y ×Y of Theorem 2.5, (i); Theorem 1.16, (iii), respectively. Moreover, α ∞ (respectively, α × ∞ ) is compatible, up cuspidally inner automorphisms, with the decomposition ≤∞ groups of x ∗ , y ∗ in Π ≤∞ U S , Π V T (respectively, with the images of the decomposition ≤∞ groups D X , D Y in Π ≤∞ U X×X , Π U Y ×Y ). Finally, this condition of “compatibility with decomposition groups”, together with the condition of making the above diagrams commute, uniquely determine the isomorphisms α ∞ , α × ∞ , up to composition with × a cuspidally inner automorphism; in particular, α ∞ is compatible, up to compo- sition with a cuspidally inner automorphism, with the automorphisms of Π ≤∞ U X×X , Π ≤∞ U Y ×Y given by switching the two factors. Proof. First, let us consider the isomorphism [i.e., more precisely: a speciﬁc mem- ber of the cuspidally inner equivalence class of isomorphisms] ∼ c-ab,l α c-ab,l : Π c-ab,l U X×X → Π U Y ×Y ABSOLUTE ANABELIAN CUSPIDALIZATIONS 75 ∼ c-ab arising from the isomorphism Π c-ab U X×X → Π U Y ×Y of Theorem 1.16, (iii). Recall that since α is Frobenius-preserving, it is quasi-point-theoretic [cf. Remark 1.18.2], and that α c-ab,l is compatible with the images of D X , D Y , which we denote by (l) (l) D X , D Y . Thus, we may assume without loss of generality that our choices of decomposition groups D x ∗ [U S ] ⊆ Π U S , D y ∗ [V T ] ⊆ Π V T , as well as our choices of c-ab,l splittings G k X → D x ∗ [U S ], G k Y → D y ∗ [V T ], have images in Π c-ab,l U X×X , Π U Y ×Y that ⊆ Π c-ab,l correspond via α c-ab,l . In particular, it follows that α c-ab,l maps Π c-ab,l U S U X×X isomorphically onto Π c-ab,l ⊆ Π c-ab,l V T U Y ×Y . In the following argument, let us identify the “Lin U S (1/∞)”, “Lin X (1/∞)” portions of Δ Lie U S with the [completions, relative to the natural ﬁltration topology, of the] corresponding graded objects “Gr Q l (−)(1/∞)” via the Galois invariant split- tings of Proposition 3.4, (ii), and similarly for V T . Then, in light of our assumption that α is Frobenius-preserving, it follows immediately from the naturality of our con- structions [cf., especially, Proposition 3.4, (iii)] that α induces, for each Z b ≥ 1, compatible isomorphisms ∼ LIE α LIE : Π LIE U S → Π V T ; ∼ α LIE≤b : Π LIE≤b → Π LIE≤b U S V T which are, moreover, compatible [with respect to the natural projections to Π X , Π Y ] ” with the isomorphism α. Moreover, it follows from the construction of “Π LIE≤b (−) that the latter displayed isomorphism maps D x LIE≤b ⊆ Π LIE≤b bijectively onto U S ∗ LIE≤b LIE≤b LIE≤b ∼ D y ∗ ⊆ Π V T , and that the resulting isomorphism D x ∗ → D y LIE≤b induces ∗ an isomorphism ∼ D x ≤b → D y ≤b ∗ ∗ which is compatible [again by construction!] with the respective Frobenius elements “F k ” on either side. Next, let us observe that since the isomorphism α c-ab,l induces an isomorphism ∼ Π c-ab,l → Π c-ab,l that is compatible with the images of the decompositions groups U S V T D x ∗ [U S ], D y ∗ [V T ] and Frobenius elements in these decomposition groups, it fol- lows immediately that for corresponding [i.e., via α] (S, ∅, Σ)-, (T, ∅, Σ)-admissible coverings X → X, Y → Y [which induce coverings U S → U S , V T → V T ], ∼ α c-ab,l induces an isomorphism Δ Lie≤2 → Δ Lie≤2 which is compatible with α LIE≤2 . V T U S , Δ Lie≤2 are not center-free, the semi-direct products Moreover, although Δ Lie≤2 V T U S Δ Lie≤2 H X , Δ Lie≤2 H Y are easily seen to be center-free [cf. Proposition 1.8, V T U S (i)], for arbitrary open subgroups H X ⊆ G † k X , H Y ⊆ G † k Y [where the daggers are as in Proposition 1.8, (i)] that correspond via α. Since Π LIE≤2 (respectively, Π LIE≤2 ) U S V T is an inverse limit of topological groups that admit normal closed subgroups of the H X (respectively, Δ Lie≤2 H Y ), we thus conclude [by applying the form Δ Lie≤2 V T U S extension “1 → (−) → Aut(−) → Out(−) → 1” of §0 to these normal closed sub- ∼ → Π c-ab,l induced by α c-ab,l is compatible — groups] that the isomorphism Π c-ab,l U S V T relative to the natural inclusions ∼ LIE≤2 Π c-ab,l → Π ≤2 ; U S → Π U S U S ∼ LIE≤2 Π c-ab,l → Π ≤2 V T → Π V T V T 76 SHINICHI MOCHIZUKI ∼ [cf. the discussion preceding Proposition 3.7] — with α LIE≤2 : Π LIE≤2 → Π LIE≤2 . U S V T In fact, since 3 is odd, it follows immediately from the deﬁnitions that the modules “New Q S (3/4)” vanish, hence [cf. Deﬁnition 3.5, (ii)] that we have an iso- LIE≤3 ∼ → Π LIE≤2+ , which implies [cf. Proposition 3.7, (i)] that we have morphism Π U S U S ≤3 ∼ ≤2 an isomorphism Π U S → Π U S [and similarly for V T ]. Thus, by Proposition 3.7, (iii), ∼ → Π c-ab,l induced by α c-ab,l is compatible — it follows that the isomorphism Π c-ab,l U S V T relative to the natural inclusions ∼ LIE≤3 → Π ≤3 ; Π c-ab,l U S U S → Π U S ∼ LIE≤3 Π c-ab,l → Π ≤3 V T V T → Π V T ∼ — with α LIE≤3 : Π LIE≤3 → Π LIE≤3 . U S V T LIE Next, let us observe that the diagonal actions of D X , D Y on Π LIE U S , Π V T clearly (l) (l) (l) (l) (l) (l) factor through D X , D Y [hence determine “diagonal actions” of D X , D Y on Π LIE U S , ]. Moreover, by what we have already shown concerning the compatibility of Π LIE V T α LIE≤3 with α c-ab,l [cf. also the compatibility of α c-ab,l with D X , D Y ] and the compatibility of α c-ab,l with the decomposition groups D x ∗ [U S ], D y ∗ [V T ], it follows [cf. Remarks 3.8.1, 3.8.2] that the conditions (a), (b), (c) of Proposition 3.8 are compatible with α LIE , hence that α LIE is compatible with the diagonal actions of (l) (l) (l) ∼ (l) LIE D X , D Y on Π LIE U S , Π V T [relative to the isomorphism D X → D Y induced by α c-ab,l ]. ≤b Now I claim that the isomorphism α LIE≤b maps Π ≤b U S bijectively onto Π V T , thus inducing a compatible inverse system [parametrized by b] of isomorphisms ∼ ≤b α ≤b : Π ≤b U S → Π V T ≤b that are compatible [with respect to the natural projections Π ≤b U S Π X , Π V T Π Y ] with α. To verify this claim, we apply induction on b. The case b = 1 is vacuous; the case b = 2 follows from what we have already shown concerning the compatibility of α LIE≤2 with α c-ab,l . Thus, we assume that b ≥ 2, and that the claim has been veriﬁed for “b” that are ≤ the b under consideration. Now observe that by Propositions 3.7, (iii); 3.9, (ii), it follows that the isomor- phism LIE≤b+1 ∼ Π U → Π V LIE≤b+1 S T maps Π ≤b+1 bijectively onto a Ker(Π LIE≤b+1 Π V LIE≤b+ )-conjugate of Π ≤b+1 . U S V T V T T LIE≤b+1 In particular, by conjugating by an appropriate element γ ∈ Ker(Π V T LIE≤b+ Π V T ), we obtain an isomorphism ∼ β b+1 : Π ≤b+1 → Π ≤b+1 U S V T that is compatible with α ≤b and, moreover, [since γ commutes with I y ≤b+1 ] maps ∗ ≤b+1 ≤b+1 bijectively onto I y ∗ . Note that by Propositions 3.7, (i); 3.9, (iii), it follows I x ∗ ABSOLUTE ANABELIAN CUSPIDALIZATIONS 77 that the choice of γ is unique, modulo Ker(Π ≤b+1 Π ≤b+ V T V T ). In particular, the TOR≤b+1 TOR≤b+1 image δ ∈ Π V T of γ in Π V T is uniquely determined. (l) On the other hand, since α LIE is compatible with the diagonal actions of D X , (l) LIE D Y on Π LIE U S , Π V T , it follows immediately, by “transport of structure”, that δ is (l) ﬁxed by the diagonal action of D Y . But, by Proposition 3.9, (iv), this implies that δ = 0. This completes the proof of the claim. ∼ ≤∞ Thus, we obtain an isomorphism α ∞ : Π ≤∞ U S → Π V T as in the statement of ≤∞ Theorem 3.10. Next, let us recall that Δ ≤∞ U S , Δ V T are slim [cf. Proposition 3.9, (v)]. Thus, since this isomorphism α ∞ is compatible with the diagonal actions of ∼ (l) (l) ≤∞ D X , D Y , we may apply the isomorphism Aut(Δ ≤∞ U S ) → Aut(Δ V T ) induced by α ∞ to obtain — i.e., by pulling back the extension ≤∞ ≤∞ 1 → Δ ≤∞ U S → Aut(Δ U S ) → Out(Δ U S ) → 1 [cf. §0] via the homomorphism (D X ) Π X → Out(Δ ≤∞ U S ) (l) × arising from the diagonal action [and similarly for Δ ≤∞ V T ] — an isomorphism α ∞ : ∼ ≤∞ Π ≤∞ U X×X → Π U Y ×Y as in the statement of Theorem 3.10. Here, we note that the “cuspidally inner indeterminacy” of α ∞ , α × ∞ that is referred to in the statement of Theorem 3.10 arises from the “cuspidally inner indeterminacy” in the choice of corresponding decomposition groups D x ∗ [U S ], D y ∗ [V T ] [more precisely: the images ≤∞ c-ab,l c-ab,l ]. Finally, we ob- of these groups in Π ≤∞ U S , Π V T , as opposed to just in Π U S , Π V T serve that the asserted uniqueness follows immediately by considering eigenspaces relative to the Frobenius actions [cf. Proposition 3.4, (ii)], together with the con- struction of the isomorphism α LIE [cf. also Propositions 1.15, (i); 2.6, (i)]. Remark 3.10.1. The argument of the proof of Theorem 3.10 involving Proposi- tion 3.9, (iv), may be regarded as a sort of “higher order analogue” of the argument applied in the proof of Theorem 1.16, (iii), involving Lemma 1.11; Proposition 1.12, (v). Remark 3.10.2. At ﬁrst glance, it may appear that the portion of Theorem 3.10 concerning α × ∞ may only be concluded when X(k X ), Y (k Y ) are nonempty. In fact, ≤∞ † † however, since (Π ≤∞ U X×X ) , (Π U Y ×Y ) are slim [cf. Proposition 3.9, (v)], it follows that the portion of Theorem 3.10 concerning α × ∞ may be concluded even without assuming that X(k X ), Y (k Y ) are nonempty, by applying Theorem 3.10 after passing to corresponding [via α] ﬁnite extensions of k X , k Y [cf. Remark 1.10.1]. Remark 3.10.3. It seems reasonable to expect that, when, say, Σ = {l}, the techniques applied in the proof of Theorem 3.10, together with the theory of [Mtm], 78 SHINICHI MOCHIZUKI should allow one to reconstruct the [geometrically pro-Σ] étale fundamental groups of the various conﬁguration spaces [i.e., ﬁnite products of copies of X over k X , with the various diagonals removed] “group-theoretically” from Π X [under, say, an appropriate hypothesis of “Frobenius-preservation” as in Theorem 3.10]. This topic, however, lies beyond the scope of the present paper. Remark 3.10.4. If the “cuspidalization of conﬁguration spaces” [cf. Remark 3.10.3] can be achieved, then it seems likely that by applying an appropriate “spe- cialization” operation, it should be possible to generalize Theorem 3.10 to the case where S, T are subsets of arbitrary ﬁnite cardinality. Remark 3.10.5. One essential portion of the proof of Theorem 3.10 is the Galois invariant splitting of Proposition 3.4, (ii). Although it does not appear likely that such a splitting exists in the case of a nonarchimedean local base ﬁeld [cf., e.g., the theory of [Mzk4]], it would be interesting to investigate the extent to which a result such as Theorem 3.10 may be generalized to the nonarchimedean local case, perhaps by making use of some sort of splitting such as the Hodge-Tate decomposition, or a splitting that arises via crystalline methods. In the context of absolute anabelian geometry over nonarchimedean local ﬁelds, however, such p-adic Hodge-theoretic splittings might not be available, since the isomorphism class of the Galois module “C p ” is not preserved by arbitrary automorphisms of the absolute Galois group of a nonarchimedean local ﬁeld [cf. the theory of [Mzk3]]. The development of the theory underlying Theorem 3.10 was motivated by the following important consequence: Corollary 3.11. (Total Global Green-compatibility) In the situation of Theorem 1.16, (iii) [in the ﬁnite ﬁeld case], suppose further that Σ † = Primes † , and that X, Y are Σ-separated [which implies that α is Frobenius-preserving and point-theoretic — cf. Remarks 1.18.1, 1.18.2]. Then the isomorphism α is totally globally Green-compatible. Proof. Indeed, we may apply Theorem 3.10 to the isomorphism α of Theorem 1.16, (iii), and arbitrary choices of sets of cardinality one S = {x ∗ }, T = {y ∗ } that correspond via α. Let l ∈ Σ † . Then let us observe that the quotient Π U S Π ≤∞ U S satisﬁes the following property: If Π U S Q is a ﬁnite quotient of Π U S such that for some quotient Q Q whose kernel has order a power of l, Π U S Q factors through Π U S ≤∞ Π ≤∞ U S , then Π U S Q also factors through Π U S Π U S . A similar statement holds for the quotient Π V T Π ≤∞ V T . In light of this observation, † † together with our assumption that Σ = Primes [which implies that α is Frobenius- preserving], it follows that the reasoning of [Tama], Corollary 2.10, Proposition 3.8 ABSOLUTE ANABELIAN CUSPIDALIZATIONS 79 [cf. also Remark 1.18.2 of the present paper], may be applied to the isomorphism ∼ ≤∞ α ∞ : Π ≤∞ U S → Π V T of Theorem 3.10 to conclude that the isomorphism α ∞ maps the set of decompo- sition subgroups of the domain bijectively onto the set of decomposition subgroups of the codomain. On the other hand, sorting through the deﬁnitions, the datum of the lifting of a decomposition group of Π X , Π Y corresponding to a point that does not belong to S, T to a [noncuspidal] decomposition group of the domain or codomain of c-ab,l α ∞ determines, by projection to Π c-ab,l , the l-adic portion of the Green’s U S , Π V T trivialization associated to this point and the unique point of S or T . Since l is an arbitrary element of Σ † = Primes † , and the points x ∗ , y ∗ are arbitrary points that correspond via α, this shows that α is globally Green-compatible. That α is totally globally Green-compatible follows by applying this argument to the isomorphism induced by α between open subgroups of Π X , Π Y . Theorem 3.12. (The Grothendieck Conjecture for Proper Hyperbolic Curves over Finite Fields) Let X, Y be proper hyperbolic curves over a ﬁnite ﬁeld; denote the base ﬁelds of X, Y by k X , k Y , respectively. Write Π X , Π Y for the étale fundamental groups of X, Y , respectively. Let ∼ α : Π X → Π Y be an isomorphism of proﬁnite groups. Then α arises from a uniquely deter- mined commutative diagram of schemes ∼ X ⏐ ⏐ → X → ∼ Y ⏐ ⏐ Y in which the horizontal arrows are isomorphisms; the vertical arrows are the pro- ﬁnite étale universal coverings determined by the proﬁnite groups Π X , Π Y . Proof. Theorem 3.12 follows formally from Corollaries 2.7, 3.11; Remarks 1.18.1, 1.18.2; Proposition 2.3, (ii). 80 SHINICHI MOCHIZUKI Appendix: Free Lie Algebras In this present Appendix, we discuss various elementary facts concerning free Lie algebras that are necessary in §3. In particular, we develop a sort of “higher order analogue” of the theory developed in Lemma 1.11. Proposition A.1. (Free Lie Algebras) Let R be a commutative ring with unity; V a ﬁnitely generated free R-module. Write Lie R (V ) for the free Lie algebra over R associated to V ; for Z b ≥ 1, denote by Lie bR (V ) ⊆ Lie R (V ) the R-submodule generated by the “alternants of degree b” [cf. [Bour], Chapter II, §2.6]. Also, we shall denote by U R (V ) the enveloping algebra of Lie R (V ). [Thus, we have a natural inclusion Lie R (V ) → U R (V ).] Then: (i) Each Lie bR (V ) is a ﬁnitely generated free R-module. Moreover, there is ∼ a natural isomorphism V → Lie 1 R (V ). (ii) Let v ∈ V be a nonzero element such that the quotient module V /R · v is free. Then the centralizer of v in U R (V ) is equal to the R-submodule of U R (V ) generated by the nonnegative powers of v. In particular, if R is a ﬁeld of characteristic zero, then the centralizer of v in Lie R (V ) is equal to R · v. (iii) Suppose that the rank of V over R is ≥ 2. Then the Lie algebra Lie R (V ) is center-free. In particular, the adjoint representation of Lie R (V ) is faithful. (iv) Let R be an R-algebra which is ﬁnitely generated and free as an R- module. Let φ : R R be a surjection of R-algebras; suppose that V = V ⊗ R ,φ R, for some ﬁnitely generated free R -module V [so we obtain a natural surjection V V compatible with φ]. Then the natural surjection V V induces a sur- jection of R-modules Lie bR (V ) Lie bR (V ) that factors as a composite of natural surjections as follows: Lie bR (V ) Lie bR (V ) Lie bR (V ) Here, the ﬁrst arrow of this factorization is the arrow naturally induced by observ- ing that every Lie algebra over R naturally determines a Lie algebra over R; the second arrow of this factorization is the arrow functorially induced by the natural φ- compatible surjection V V . Finally, this second arrow induces an isomorphism ∼ Lie bR (V ) ⊗ R ,φ R → Lie bR (V ). Proof. Assertion (i) follows immediately from [Bour], Chapter II, §2.11, Theorem 1, Corollary. Assertion (ii) follows from the well-known structure of the enveloping algebra U R (V ) [i.e., the natural isomorphism of U R (V ) with the free associative algebra determined by V over R; the fact that when R is a ﬁeld of characteristic zero, the image of Lie R (V ) in U R (V ) may be identiﬁed with the set of primitive ele- ments — cf. [Bour], Chapter II, §3, Theorem 1, Corollaries 1,2], by considering the eﬀect on “words” of forming the commutator with v — cf. the argument of [Mtm], Proposition 3.1 [which is given only in the case where R is a ﬁeld of characteristic ABSOLUTE ANABELIAN CUSPIDALIZATIONS 81 zero, but does not, in fact, make use of this assumption on R in an essential way]. Assertion (iii) follows immediately from assertion (ii) [by allowing the element “v” of assertion (ii) to range over the elements of an R-basis of V ]. Assertion (iv) fol- lows formally from the universal property of a free Lie algebra, together with the well-known functoriality of a free Lie algebra with respect to tensor products [cf. [Bour], Chapter II, §2.5, Proposition 3]. Next, let us suppose that we have been given data as follows: (i) a prime number l; (ii) a proﬁnite group Δ that admits an normal open subgroup V ⊆ Δ such that the following conditions are satisﬁed: (a) V is abelian [so we shall regard V as a module]; (b) the topological module V is a ﬁnitely generated def free R-module, where we write R = Z l ; (c) the resulting action of the def def ﬁnite group G = Δ/V on V determines a G-module V Q l = V ⊗ Q l that contains the regular representation of G; (iii) a positive power N of l; (iv) a collection of [not necessarily distinct!] elements g 1 , . . . , g d ∈ G [where d ≥ 1 is an integer] of G at least one of which is not equal to the identity element. Write def ζ = d (1 − g i ) ∈ R[G] i=1 [where R[G] is the group ring of G with coeﬃcients in R]. Then we have the following result: Lemma A.2. (Nontriviality of a Certain Operator) There exists an integer n ≥ 1 such that the order |J ζ | of the image J ζ ⊆ J def of the action of ζ on [the ﬁnite group] J = V ⊗ (Z/l n Z) is divisible by N . Proof. Indeed, since the G-module V Q l contains the regular representation [cf. condition (ii), (c)], it follows that the image of the action of ζ on V Q l is a nonzero Q l -vector space, hence that the image of the action of ζ on the ﬁnitely generated free R-module V [cf. condition (ii), (b)] contains a rank one free R-module. Now Lemma A.2 follows immediately. def Next, let J ζ ⊆ J be as in Lemma A.2; write H = Δ/(l n · V ) [so J ⊆ H, H/J = G]. Also, let us assume that we have been given data as follows: 82 SHINICHI MOCHIZUKI (v) a collection of elements h 1 , . . . , h d ∈ H that lift g 1 , . . . , g d ∈ G; (vi) a G-torsor E G [whose G-action will be written as an action from the left]; (vii) an H-torsor E H [whose H-action will be written as an action from the left]; (viii) a surjection : E H E G that is compatible with the natural surjection H G; (ix) a continuous action of a proﬁnite group Γ on Δ that preserves the sub- def group V ⊆ Δ, hence determines a proﬁnite group Δ Γ = Δ Γ that acts continuously on G, H [in such a way that the restriction of this action to Δ ⊆ Δ Γ is the action of Δ on G, H by conjugation]; (x) continuous actions of Δ Γ on E G , E H [which will be denoted via super- scripts] that are compatible with the continuous actions of Δ Γ on G, H, as well as with the surjection and, moreover, induce the trivial action of Γ ⊆ Δ Γ on E G [hence also on G]; (xi) an element [i.e., “basepoint”] e H ∈ E H , whose image via we denote by e G ∈ E G , such that e H , e G are ﬁxed by the action of Δ Γ on E H , E G . Next, let us write def R J = R[J ] for the group ring of J with coeﬃcients in R. Thus, R J is a commutative R-algebra, and we have a natural augmentation homomorphism R J R [which sends all of the elements of J to 1]. Moreover, if we write def def M : M H = R[E H ] M G = R[E G ] for the morphism of R J -modules induced by on the respective free R-modules with bases given by the elements of E H , E G , then M induces a natural isomorphism ∼ M H ⊗ R J R → M G . Thus, it follows from Proposition A.1, (iv), that, for b ≥ 1 an integer, we have [in the notation of Proposition A.1] natural surjections Lie bR (M H ) Lie bR J (M H ) Lie bR (M G ) ∼ the second of which determines a natural isomorphism Lie bR J (M H )⊗ R J R → Lie bR (M G ). Now let P (X 1 , . . . , X d ) be an “alternant monomial of degree d” [i.e., a monomial element of Lie d Z (−) of the free Z-module on the indeterminate symbols X 1 , . . . , X d ] in which each X i [for i = 1, . . . , d] appears precisely once. Then P (X 1 , . . . , X d ) determines an element P (g 1 · e G , . . . , g i · e G , . . . , g d · e G ) ABSOLUTE ANABELIAN CUSPIDALIZATIONS 83 of Lie dR (M G ). Moreover, by allowing such P (X 1 , . . . , X d ) and g 1 , . . . , g d to vary ap- propriately, we obtain a Hall basis [cf., e.g., [Bour], Chapter II, §2.11] of Lie dR (M G ) [at least if d ≥ 2; if d = 1, then one must also allow for the unique g 1 to be the identity element]. Similarly, by allowing such P (X 1 , . . . , X d ) and h 1 , . . . , h d ∈ H to vary appropriately, we obtain a Hall basis [again, strictly speaking, if d ≥ 2] of Lie dR J (M H ) of elements of the form P (h 1 · e H , . . . , h d · e H ). Lemma A.3. (Relation of Superscript and Left Actions) For any v ∈ V ⊆ Δ ⊆ Δ Γ that maps to j ∈ J, we have P (h 1 · e H , . . . , h i · e H , . . . , h d · e H ) v = ζ(j) · P (h 1 · e H , . . . , h i · e H , . . . , h d · e H ) in Lie dR J (M H ). Proof. Indeed, we compute: P (h 1 · e H , . . . ,h i · e H , . . . , h d · e H ) v = P (h v 1 · e H , . . . , h vi · e H , . . . , h vd · e H ) −1 −1 v v = P (h v 1 · h −1 1 · h 1 · e H , . . . , h i · h i · h i · e H , . . . , h d · h d · h d · e H ) = d [j, h i ] · P (h 1 · e H , . . . , h i · e H , . . . , h d · e H ) i=1 = ζ(j) · P (h 1 · e H , . . . , h i · e H , . . . , h d · e H ) [where we apply the R J -module structure of E H and the fact that e vH = e H [cf. (xi)]]. Next, let us assume that we have also been given the following data: (xii) a topological R-module Λ equipped with a continuous action by Γ [which thus determines, via the natural surjection Δ Γ Γ, a continuous action by Δ Γ on Λ]. Write: def V Γ = V Γ ⊆ Δ Γ ; def F J = J · P (h 1 · e H , . . . , h d · e H ) ⊆ Lie dR J (M H ); def R[F J ] = R · F J = R J · P (h 1 · e H , . . . , h d · e H ) ⊆ Lie dR J (M H ); def Λ[F J ] = R[F J ] ⊗ R Λ ⊆ Lie dR J (M H )⊗ R ; Λ def F = P (g 1 · e G , . . . , g d · e G ) ∈ Lie dR (M G ); def R[F ] = R · F ⊆ Lie dR (M G ); def Λ[F ] = R[F ] ⊗ R Λ ⊆ Lie dR (M G ) ⊗ R Λ Thus, the natural surjection Lie dR J (M H ) Lie dR (M G ) determines [compatible] nat- ural surjections F J {F }, R[F J ] R[F ], Λ[F J ] Λ[F ]. Also, we observe [cf. the 84 SHINICHI MOCHIZUKI fact that Lie dR J (M H ) is a ﬁnitely generated free R J -module] that F J is a J-torsor [relative to the action from the left], hence, in particular, a ﬁnite set. Now observe that since V Γ acts trivially on G, e H [cf. (ix), (x), (xi)], it follows immediately that V Γ acts compatibly on F J , R[F J ], Λ[F J ], F , R[F ], Λ[F ], and that the natural action of V Γ on R[G] preserves ζ. In particular, it follows that V Γ preserves J ζ ⊆ J, hence that V Γ acts naturally on the set of orbits (F J ) F ζ of F J with respect to the action of J ζ ; moreover, by Lemma A.3, it follows that this action of V Γ on F ζ factors through the quotient V Γ Γ. Now let us consider invariants with respect to the various superscript actions under consideration. Let us write Invar(−, −) for the set of invariants of the second argument in parentheses with respect to the superscript action of the group given by the ﬁrst argument in parentheses. Then any element η ∈ Invar(V Γ , Λ[F J ]) may be regarded as a Λ-valued function on the set F J which descends [cf. Lemma A.3] to a Γ-invariant Λ-valued function on F ζ , i.e., an element η ζ ∈ Invar(Γ, Λ[F ζ ]). Next, let us observe that [since η ζ is Γ-invariant] the sum of the values ∈ Λ of the Λ- valued function on F ζ determined by η ζ is a Γ-invariant element η ζ ∈ Invar(Γ, Λ). Thus, the sum η ∈ Λ of the values ∈ Λ of the Λ-valued function on F J determined by η satisﬁes the relation η ζ η = |J ζ | · in Λ. But the image of η in Λ[F ] is precisely the element ( η) · F . Thus, since, by Lemma A.2, |J ζ | is divisible by N , we conclude the following: Lemma A.4. (Monomial-wise Computation of Invariants) The image Im(Invar(V Γ , Λ[F J ])) ⊆ Λ[F ] of Invar(V Γ , Λ[F J ]) ⊆ Λ[F J ] in Λ[F ] lies in N · Invar(Γ, Λ[F ]). Thus, by allowing P (X 1 , . . . , X d ) and h 1 , . . . , h d ∈ H as in the above discus- sion to vary appropriately so as to obtain a Hall basis [again, strictly speaking, if d ≥ 2] of Lie dR J (M H ) of elements of the form P (h 1 · e H , . . . , h d · e H ), we conclude the following: ABSOLUTE ANABELIAN CUSPIDALIZATIONS 85 Theorem A.5. (Invariants of Free Lie Algebras) Let d ≥ 1 be an integer. Suppose that we have been given data as in (i), (ii), (iii) above. Let n ≥ 1 be an integer that satisﬁes the property of Lemma A.2 for all [of the ﬁnitely many] def possible choices of data as in (iv) [relative to the given integer d ≥ 1]; J = V /(l n · def def V ) ⊆ H = Δ/(l n · V ); R J = R[J ]. Suppose that have also been given data as def def in (vi), (vii), (viii), (ix), (x), (xi), (xii) above; let M H = R[E H ], M G = R[E G ], def V Γ = V Γ (⊆ Δ Γ ). Then the natural surjection Lie dR J (M H ) ⊗ R Λ Lie dR (M G ) ⊗ R Λ maps Invar(V Γ , Lie dR J (M H ) ⊗ R Λ) into N · Invar(V Γ , Lie dR (M G ) ⊗ R Λ) if d ≥ 2. In a similar vein, the natural surjection M H ⊗ R Λ M G ⊗ R Λ maps Invar(V Γ , M H ⊗ R Λ) into N · Invar(V Γ , M G ⊗ R Λ) + Invar(V Γ , Λ) · e G ⊆ M G ⊗ R Λ. 86 SHINICHI MOCHIZUKI Bibliography [Bour] N. Bourbaki, Lie Groups and Lie Algebras, Springer Verlag (1989). [DM] P. Deligne and D. Mumford, The Irreducibility of the Moduli Space of Curves of Given Genus, IHES Publ. Math. 36 (1969), pp. 75-109. [FC] G. Faltings and C.-L. 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